<div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w0 h0" alt="" src="https://files.passeidireto.com/901d6444-6f43-4885-b13e-258399dbb0ec/bg1.png"><div class="t m0 x1 h1 y1 ff1 fs0 fc0 sc0 ls1 ws0"> <span class="_0 blank"> </span> </div><div class="t m0 x2 h2 y2 ff1 fs1 fc0 sc0 ls1 ws0"> <span class="_1 blank"> </span> </div><div class="t m0 x3 h3 y3 ff1 fs1 fc0 sc0 ls0 ws0"> <span class="fs0 ls1 v1"> </span></div><div class="t m1 x4 h2 y4 ff1 fs1 fc0 sc0 ls1 ws0"> </div><div class="t m0 x5 h1 y5 ff1 fs0 fc0 sc0 ls1 ws0"> </div><div class="t m1 x6 h2 y6 ff1 fs1 fc0 sc0 ls1 ws0"> </div><div class="t m0 x7 h4 y7 ff2 fs2 fc1 sc0 ls1 ws0">Ensino <span class="_2 blank"></span>Superior </div><div class="t m0 x8 h1 y8 ff2 fs0 fc2 sc0 ls1 ws0">1.2. <span class="_2 blank"></span>Integral Indefinida </div><div class="t m0 x9 h2 y9 ff2 fs1 fc2 sc0 ls1 ws0"> <span class="_2 blank"></span> Integração por Partes </div><div class="t m0 xa h2 ya ff1 fs1 fc2 sc0 ls1 ws0">Amintas P<span class="_3 blank"></span>aiv<span class="_2 blank"></span>a Afonso </div><div class="t m0 x5 h1 yb ff1 fs0 fc0 sc0 ls1 ws0"> </div><div class="t m0 x5 h2 yc ff1 fs1 fc0 sc0 ls1 ws0"> </div><div class="t m0 xb h5 yd ff2 fs3 fc2 sc0 ls1 ws0">Cálculo 2 </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w0 h0" alt="" src="https://files.passeidireto.com/901d6444-6f43-4885-b13e-258399dbb0ec/bg2.png"><div class="t m0 xc h6 ye ff3 fs4 fc3 sc0 ls1 ws0"> <span class="ls2 ws1">Técnicas</span><span class="ls3"> <span class="ls2 ws1">de</span></span> Integração<span class="_4 blank"></span> </div><div class="t m0 xd h6 yf ff4 fs5 fc4 sc0 ls4">\uf06e<span class="ff3 fs4 fc5 ls1 ws0">Integração <span class="_4 blank"> </span><span class="ls2 ws1">por</span> partes<span class="_4 blank"></span><span class="ff5">: </span></span></div><div class="t m0 xe h7 y10 ff5 fs4 fc0 sc0 ls5 ws0"> <span class="ls11 ws2">No</span><span class="ls1"> <span class="_4 blank"> </span>Cálc<span class="_4 blank"></span>ulo <span class="_4 blank"> </span>1, <span class="_4 blank"> </span><span class="ls6 ws3">quando<span class="_4 blank"></span></span> <span class="_4 blank"> </span>calcul<span class="_4 blank"></span>ávamos<span class="_4 blank"></span> <span class="_4 blank"> </span>a <span class="_4 blank"> </span><span class="ff6">derivada<span class="_4 blank"> </span> <span class="_4 blank"> </span><span class="ls2 ws1">do</span> <span class="_5 blank"> </span>produto<span class="_4 blank"> </span></span><span class="ls7"> <span class="ls6 ws3">de</span></span> </span></div><div class="t m0 xf h7 y11 ff5 fs4 fc0 sc0 ls6 ws3">duas<span class="ls1 ws0"> <span class="_6 blank"> </span>funções<span class="_4 blank"></span> <span class="_6 blank"> </span>aplicávamos <span class="_6 blank"> </span>uma <span class="_6 blank"> </span>regra<span class="_4 blank"> </span>: <span class="_6 blank"> </span>chamá<span class="_4 blank"></span>vamos <span class="_6 blank"> </span>u<span class="_4 blank"></span>ma <span class="_6 blank"> </span></span>das<span class="ls1 ws0"> </span></div><div class="t m0 xf h8 y12 ff5 fs4 fc0 sc0 ls1 ws0">funções<span class="_4 blank"></span> <span class="_7 blank"> </span><span class="ls6 ws3">de</span><span class="ls8"> </span><span class="ff7 fs1 ws4">u</span>, <span class="_7 blank"> </span>a <span class="_7 blank"> </span>outra<span class="_4 blank"></span> <span class="_7 blank"> </span>funç<span class="_4 blank"></span>ão <span class="_7 blank"> </span><span class="ls6 ws3">de</span><span class="ls9"> <span class="ff7 fs1 lsa">v</span></span> <span class="_7 blank"> </span>e<span class="_4 blank"></span> <span class="_7 blank"> </span>sua <span class="_7 blank"> </span>deriv<span class="_4 blank"></span>ada <span class="_7 blank"> </span>era<span class="_4 blank"> </span> <span class="_7 blank"> </span><span class="ls6 ws3">dada</span><span class="ls8"> <span class="ls6 ws3">por</span></span> </div><div class="t m0 xf h9 y13 ff8 fs1 fc0 sc0 ls1 ws5">u\u2019v<span class="ff9 ws0"> + </span>uv\u2019<span class="ff5 fs4 ws0">. </span></div><div class="t m0 xd h8 y14 ff4 fs5 fc4 sc0 ls4">\uf06e<span class="ff5 fs4 fc5 ls1 ws0"> Exemplo: <span class="fc0">Sej<span class="_4 blank"></span>a <span class="ff7 fs1">f(x) = e<span class="fs6 v2">x </span>. senx</span>. Chamamos <span class="_4 blank"> </span><span class="ff7 fs1">u = e<span class="fs6 lsb v2">x</span></span><span class="lsc">,<span class="ls12"> </span></span><span class="ff7 fs1">v = senx</span><span class="ls12"> </span>e </span></span></div><div class="t m0 xd ha y15 ff5 fs4 fc0 sc0 ls1 ws0"> <span class="_4 blank"> </span><span class="ff8 fs1 ws5">f\u2019(x)<span class="_4 blank"> </span><span class="ff7 ws0"> = e<span class="fs6 v2">x </span>. senx<span class="ffa"> </span>+ e<span class="fs6 v2">x </span>. cosx<span class="ff5"> </span></span></span></div><div class="t m0 xd h7 y16 ff4 fs5 fc4 sc0 ls4">\uf06e<span class="ff5 fs4 fc0 ls1 ws0">A <span class="_8 blank"> </span>integração<span class="_4 blank"> </span> <span class="_8 blank"> </span><span class="ls6 ws3">por</span> <span class="_8 blank"> </span>parte<span class="_4 blank"> </span>s <span class="_8 blank"> </span>irá<span class="_4 blank"></span> <span class="_8 blank"> </span><span class="lsb ws6">se</span> <span class="_8 blank"> </span>aplicar<span class="_4 blank"></span> <span class="_8 blank"> </span>a <span class="_8 blank"> </span>esses<span class="_4 blank"></span> <span class="_8 blank"> </span>casos<span class="_4 blank"></span> <span class="_8 blank"> </span><span class="ls6 ws3">em</span><span class="lsd"> <span class="ls6 ws3">que</span></span> <span class="_8 blank"> </span>a </span></div><div class="t m0 x10 hb y17 ff5 fs7 fc0 sc0 ls1 ws0">função <span class="_9 blank"> </span>é <span class="_7 blank"> </span>co<span class="_4 blank"> </span>nstituída<span class="_4 blank"> </span> <span class="_9 blank"> </span><span class="lse ws7">por</span><span class="lsf"> <span class="ls13 ws8">um</span></span> <span class="_7 blank"> </span>produto<span class="_4 blank"> </span> <span class="_9 blank"> </span>e <span class="_7 blank"> </span>ta<span class="_4 blank"> </span>mbém <span class="_9 blank"> </span><span class="lse ws7">nos</span> <span class="_9 blank"> </span>casos<span class="_4 blank"></span> <span class="_7 blank"> </span><span class="lse ws7">em</span><span class="lsf"> <span class="lse ws7">que</span></span> </div><div class="t m0 x10 h7 y18 ff5 fs4 fc0 sc0 ls1 ws0">uma <span class="ls6 ws3">das</span> funções<span class="_4 blank"> </span> <span class="ls6 ws3">pode</span> ser <span class="_4 blank"> </span>derivada <span class="_4 blank"> </span>repetidamente<span class="_4 blank"> </span> <span class="_4 blank"> </span>e a outra<span class="_4 blank"> </span> </div><div class="t m0 xd h7 y19 ff5 fs4 fc0 sc0 ls10 ws0"> <span class="ls6 ws3">pode</span><span class="ls1"> ser integrad<span class="_4 blank"></span>a repeti<span class="_4 blank"></span>damente.<span class="fc5"> </span></span></div><div class="t m0 x11 hc y1a ff1 fs8 fc2 sc0 ls1 ws0">Integr<span class="_3 blank"></span>al <span class="_3 blank"></span>Indefinida </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w0 h0" alt="" src="https://files.passeidireto.com/901d6444-6f43-4885-b13e-258399dbb0ec/bg3.png"><div class="t m0 x12 h9 y1b ff4 fs9 fc3 sc0 ls14">\uf06e<span class="ff5 fs1 fc6 ls1 ws0"> <span class="ff3 fc3">Técnicas de Integração<span class="_2 blank"></span> </span></span></div><div class="t m0 x13 ha y1c ff4 fsa fc4 sc0 ls15">\uf06e<span class="ff5 fs1 fc5 ls1 ws0">Integração por partes: </span></div><div class="t m0 x14 hd y1d ff5 fs0 fc0 sc0 ls1 ws0">Assim, considere <span class="ff7 fsb ws9">f(x)</span> <span class="ls18">e </span><span class="ff7 fsb ws9">g(x)</span> duas funções deriváveis. </div><div class="t m0 x15 ha y1e ff5 fs1 fc0 sc0 ls1 ws0">A regra do produto nos diz <span class="_4 blank"> </span>que: </div><div class="t m0 x14 ha y1f ff5 fs1 fc0 sc0 ls1 ws0"> </div><div class="t m0 x14 he y20 ff5 fs0 fc0 sc0 ls1 ws0"> </div><div class="t m0 x16 ha y21 ff5 fs1 fc0 sc0 ls1 ws0"> </div><div class="t m0 x13 ha y22 ff4 fsa fc4 sc0 ls15">\uf06e<span class="ff5 fs1 fc0 ls1 ws0">Ou, dito de outra maneira:<span class="fc5"> </span></span></div><div class="c x17 y23 w1 hf"><div class="t m2 x18 h10 y24 ffb fsc fc0 sc0 ls1 wsa">\uf05b \uf05d</div><div class="t m3 x19 h11 y25 ffc fsd fc0 sc0 ls1 wsb">)<span class="_a blank"></span>(<span class="_b blank"></span>'<span class="_c blank"></span><span class="ls19 wsc">).<span class="_d blank"></span><span class="ls1 wsb">(<span class="_e blank"></span>)<span class="_a blank"></span>(<span class="_f blank"></span><span class="ls19 wsc">).<span class="_d blank"></span><span class="ls1 wsb">(<span class="_b blank"></span>'<span class="_10 blank"></span>)<span class="_a blank"></span>(<span class="_f blank"></span><span class="ls19 wsc">).<span class="_d blank"></span><span class="ls16">(<span class="ffa ls1 wsb">x<span class="_11 blank"></span>g<span class="_12 blank"></span>x<span class="_13 blank"></span>f<span class="_14 blank"></span>x<span class="_15 blank"></span>g<span class="_12 blank"></span>x<span class="_16 blank"></span>f<span class="_17 blank"></span>x<span class="_15 blank"></span>g<span class="_12 blank"></span>x<span class="_13 blank"></span>f</span></span></span></span></span></span></span></div><div class="t m3 x1a h12 y26 ffa fsd fc0 sc0 ls1a">dx</div><div class="t m3 x1b h12 y27 ffa fsd fc0 sc0 ls17">d<span class="ffb ls1 wsd v3">\uf02b<span class="_18 blank"></span>\uf03d</span></div></div><div class="c x1c y28 w2 h13"><div class="t m4 x1d h14 y29 ffb fse fc0 sc0 ls1 wse">\uf05b \uf05d</div><div class="t m5 x1e h15 y2a ffc fsf fc0 sc0 ls1 wsf">'<span class="_19 blank"></span><span class="ls1b ws10">'.<span class="_1a blank"></span><span class="ls1 ws11">'<span class="_1b blank"></span>. <span class="ffa ls1c ws12">uv<span class="_1c blank"></span><span class="ls1 ws13">v<span class="_1d blank"></span>u<span class="_1e blank"></span>v<span class="_1f blank"></span>u <span class="ffb ws14">\uf02b<span class="_19 blank"></span>\uf03d</span></span></span></span></span></div></div><div class="t m0 x1f hc y1a ff1 fs8 fc2 sc0 ls1 ws0">Integr<span class="_3 blank"></span>al <span class="_3 blank"></span>Indefinida </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w0 h0" alt="" src="https://files.passeidireto.com/901d6444-6f43-4885-b13e-258399dbb0ec/bg4.png"><div class="t m0 x20 h16 y2b ff4 fs10 fc4 sc0 ls1d">\uf06e<span class="ff5 fs11 fc0 ls1 ws0">Em termos <span class="_3 blank"></span>de integrais <span class="_2 blank"></span>indefinidas, <span class="_2 blank"></span>a equação <span class="_3 blank"></span>se torna:<span class="_4 blank"> </span><span class="fc5"> </span></span></div><div class="t m0 xd h16 y2c ff5 fs11 fc5 sc0 ls1 ws0"> </div><div class="c x21 y2d w3 h17"><div class="t m6 x22 h18 y2e ffb fs12 fc0 sc0 ls1 ws15">\uf05b \uf05d</div><div class="t m7 x23 h19 y2f ffc fs13 fc0 sc0 ls1 ws16">)<span class="_20 blank"></span>(<span class="_21 blank"></span>'<span class="_22 blank"></span><span class="ls25 ws17">).<span class="_23 blank"></span><span class="ls1 ws16">(<span class="_24 blank"></span>)<span class="_20 blank"></span>(<span class="_1f blank"></span><span class="ls25 ws17">).<span class="_23 blank"></span><span class="ls1 ws16">(<span class="_21 blank"></span>'<span class="_14 blank"></span>)<span class="_20 blank"></span>(<span class="_1f blank"></span><span class="ls25 ws17">).<span class="_23 blank"></span><span class="ls1e">(<span class="ffa ls1 ws16">x<span class="_25 blank"></span>g<span class="_13 blank"></span>x<span class="_a blank"></span>f<span class="_26 blank"></span>x<span class="_1b blank"></span>g<span class="_27 blank"></span>x<span class="_15 blank"></span>f<span class="_28 blank"></span>x<span class="_29 blank"></span>g<span class="_27 blank"></span>x<span class="_2a blank"></span>f</span></span></span></span></span></span></span></div><div class="t m7 x1a h1a y30 ffa fs13 fc0 sc0 ls26">dx</div><div class="t m7 x24 h1a y31 ffa fs13 fc0 sc0 ls1f">d<span class="ffb ls1 ws18 v4">\uf02b<span class="_2b blank"></span>\uf03d</span></div></div><div class="c x25 y32 w4 h1b"><div class="t m8 x26 h1c y33 ffb fs14 fc0 sc0 ls1 ws19">\uf05b<span class="_2c blank"> </span>\uf05d \uf05b<span class="_2d blank"> </span>\uf05d</div><div class="t m9 x27 h1d y34 ffa fs15 fc0 sc0 ls27 ws1a">dx<span class="_2e blank"></span><span class="ls1 ws1b">x<span class="_c blank"></span>g<span class="_29 blank"></span>x<span class="_2f blank"></span>f<span class="_30 blank"></span>x<span class="_31 blank"></span>g<span class="_29 blank"></span>x<span class="_32 blank"></span>f<span class="_33 blank"></span><span class="ls27 ws1a">dx<span class="_2e blank"></span><span class="ls1 ws1b">x<span class="_31 blank"></span>g<span class="_29 blank"></span>x<span class="_2f blank"></span>f</span></span></span></div><div class="t m9 x12 h1d y35 ffa fs15 fc0 sc0 ls27">dx</div><div class="t m9 x7 h1d y36 ffa fs15 fc0 sc0 ls20">d<span class="ffc ls1 ws0 v5"> <span class="_34 blank"></span>)<span class="_35 blank"></span>(<span class="_36 blank"></span>'<span class="_37 blank"></span><span class="ls28 ws1c">).<span class="_38 blank"></span><span class="ls1 ws1b">(<span class="_39 blank"></span>)<span class="_35 blank"></span>(<span class="_a blank"></span><span class="ls28 ws1c">).<span class="_38 blank"></span><span class="ls1 ws0">(<span class="_36 blank"></span>'<span class="_3a blank"></span> <span class="_34 blank"></span>)<span class="_35 blank"></span>(<span class="_a blank"></span><span class="ls28 ws1c">).<span class="_38 blank"></span><span class="ls21">(<span class="ffb fs16 ls1 ws1d v6">\uf0f2<span class="_3b blank"></span>\uf0f2 <span class="fs15 ws1e v7">\uf02b<span class="_3c blank"></span>\uf03d</span></span></span></span></span></span></span></span></span></div></div><div class="c x28 y37 w5 h1b"><div class="t ma x26 h1c y33 ffb fs14 fc0 sc0 ls1 ws1f">\uf05b \uf05d</div><div class="t mb x29 h1d y34 ffa fs15 fc0 sc0 ls27 ws1a">dx<span class="_3d blank"></span><span class="ls1 ws1b">x<span class="_c blank"></span>g<span class="_29 blank"></span>x<span class="_2f blank"></span>f<span class="_3e blank"></span><span class="ls27 ws1a">dx<span class="_3d blank"></span><span class="ls1 ws1b">x<span class="_31 blank"></span>g<span class="_29 blank"></span>x<span class="_32 blank"></span>f<span class="_3f blank"></span><span class="ls27 ws1a">dx<span class="_2e blank"></span><span class="ls1 ws1b">x<span class="_31 blank"></span>g<span class="_29 blank"></span>x<span class="_2f blank"></span>f</span></span></span></span></span></div><div class="t mb x12 h1d y35 ffa fs15 fc0 sc0 ls27">dx</div><div class="t mb x7 h1d y36 ffa fs15 fc0 sc0 ls22">d<span class="ffc ls1 ws0 v5"> <span class="_40 blank"></span>)<span class="_35 blank"></span>(<span class="_36 blank"></span>'<span class="_37 blank"></span><span class="ls28 ws1c">).<span class="_38 blank"></span><span class="ls1 ws0">(<span class="_41 blank"></span> <span class="_40 blank"></span>)<span class="_35 blank"></span>(<span class="_a blank"></span><span class="ls28 ws1c">).<span class="_38 blank"></span><span class="ls1 ws0">(<span class="_36 blank"></span>'<span class="_42 blank"></span> <span class="_34 blank"></span>)<span class="_35 blank"></span>(<span class="_a blank"></span><span class="ls28 ws1c">).<span class="_38 blank"></span><span class="ls23">(<span class="ffb fs16 ls1 ws20 v6">\uf0f2 \uf0f2<span class="_43 blank"></span>\uf0f2<span class="_44 blank"> </span><span class="fs15 ws1e v7">\uf02b<span class="_45 blank"></span>\uf03d</span></span></span></span></span></span></span></span></span></div></div><div class="c x28 y38 w5 h1b"><div class="t ma x26 h1c y33 ffb fs14 fc0 sc0 ls1 ws1f">\uf05b \uf05d</div><div class="t mb x2a h1e y39 ffb fs16 fc0 sc0 ls1 ws21">\uf0f2<span class="_46 blank"></span>\uf0f2<span class="_47 blank"></span>\uf0f2 <span class="fs15 ws22 v7">\uf03d<span class="_45 blank"></span>\uf02d <span class="ffa ls27 ws1a">dx<span class="_3d blank"></span><span class="ls1 ws1b">x<span class="_c blank"></span>g<span class="_29 blank"></span>x<span class="_2f blank"></span>f<span class="_3f blank"></span><span class="ls27 ws1a">dx<span class="_3d blank"></span><span class="ls1 ws1b">x<span class="_31 blank"></span>g<span class="_29 blank"></span>x<span class="_32 blank"></span>f<span class="_48 blank"></span><span class="ls27 ws1a">dx<span class="_2e blank"></span><span class="ls1 ws1b">x<span class="_31 blank"></span>g<span class="_29 blank"></span>x<span class="_2f blank"></span>f</span></span></span></span></span></span></span></div><div class="t mb x12 h1d y35 ffa fs15 fc0 sc0 ls27">dx</div><div class="t mb x7 h1d y36 ffa fs15 fc0 sc0 ls24">d<span class="ffc ls1 ws0 v5"> <span class="_40 blank"></span>)<span class="_35 blank"></span>(<span class="_36 blank"></span>'<span class="_37 blank"></span><span class="ls28 ws1c">).<span class="_38 blank"></span><span class="ls1 ws0">(<span class="_49 blank"></span> <span class="_40 blank"></span>)<span class="_35 blank"></span>(<span class="_a blank"></span><span class="ls28 ws1c">).<span class="_38 blank"></span><span class="ls1 ws0">(<span class="_36 blank"></span>'<span class="_4a blank"></span> <span class="_34 blank"></span>)<span class="_35 blank"></span>(<span class="_a blank"></span><span class="ls28 ws1c">).<span class="_38 blank"></span><span class="ls1">(</span></span></span></span></span></span></span></div></div><div class="t m0 x2b h2 y3a ff1 fs1 fc0 sc0 ls1 ws0"> </div><div class="t m0 x2c hc y1a ff1 fs8 fc2 sc0 ls1 ws0">Integr<span class="_3 blank"></span>al <span class="_3 blank"></span>Indefinida </div><div class="t m0 x2d h2 y3b ff1 fs1 fc0 sc0 ls1 ws0"> </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf5" class="pf w0 h0" data-page-no="5"><div class="pc pc5 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w0 h0" alt="" src="https://files.passeidireto.com/901d6444-6f43-4885-b13e-258399dbb0ec/bg5.png"><div class="t m0 x20 h1f y3c ff4 fs17 fc4 sc0 ls29">\uf06e<span class="ff5 fs18 fc0 ls1 ws0">Em termos <span class="_3 blank"></span>de integrais <span class="_3 blank"></span>ind<span class="_4 blank"></span>efinidas, <span class="_3 blank"></span>a equação <span class="_3 blank"></span>se torna:<span class="_4 blank"></span><span class="fc5"> </span></span></div><div class="t m0 xd h16 y3d ff5 fs11 fc5 sc0 ls1 ws0"> </div><div class="c x2e y3e w6 h20"><div class="t mc x2f h21 y3f ffc fs19 fc0 sc0 ls1 ws23">'<span class="_4b blank"></span>.<span class="_4c blank"></span><span class="ls2f ws24">'.<span class="_4d blank"></span><span class="ls1 ws25">)<span class="_4e blank"></span>.<span class="_4f blank"></span>( <span class="ffa ws23">v<span class="_50 blank"></span>u<span class="_3d blank"></span>v<span class="_51 blank"></span>u<span class="_24 blank"></span>v<span class="_50 blank"></span>u</span></span></span></div><div class="t mc x1a h22 y40 ffa fs19 fc0 sc0 ls30">dx</div><div class="t mc x24 h22 y41 ffa fs19 fc0 sc0 ls2a">d<span class="ffb ls1 ws26 v8">\uf02b<span class="_52 blank"></span>\uf03d</span></div></div><div class="c x30 y42 w7 h1b"><div class="t md x31 h1d y34 ffa fs15 fc0 sc0 ls2b ws27">dx<span class="_39 blank"></span><span class="ls1 ws1b">v<span class="_53 blank"></span>u<span class="_f blank"></span>v<span class="_54 blank"></span>u<span class="_55 blank"></span><span class="ls2b ws27">dx<span class="_56 blank"></span>uv</span></span></div><div class="t md x12 h1d y35 ffa fs15 fc0 sc0 ls2b">dx</div><div class="t md x7 h1d y36 ffa fs15 fc0 sc0 ls2c">d<span class="ffc ls1 ws0 v5"> <span class="_40 blank"></span>)<span class="_36 blank"></span>'<span class="_57 blank"></span>.<span class="_58 blank"></span><span class="ls31 ws28">'.<span class="_50 blank"></span><span class="ls1 ws0">(<span class="_52 blank"></span> <span class="_40 blank"></span>)<span class="_1f blank"></span>(<span class="_59 blank"> </span><span class="ffb fs16 ws29 v6">\uf0f2<span class="_5a blank"></span>\uf0f2 <span class="fs15 ws1e v7">\uf02b<span class="_41 blank"></span>\uf03d</span></span></span></span></span></div></div><div class="c x32 y43 w8 h1b"><div class="t me x33 h1d y34 ffa fs15 fc0 sc0 ls2b ws27">dx<span class="_f blank"></span><span class="ls1 ws1b">v<span class="_53 blank"></span>u<span class="_5b blank"></span><span class="ls2b ws27">dx<span class="_27 blank"></span><span class="ls1 ws1b">v<span class="_54 blank"></span>u<span class="_5c blank"></span><span class="ls2b ws27">dx<span class="_3d blank"></span><span class="ls1 ws1b">v<span class="_5d blank"></span>u</span></span></span></span></span></div><div class="t me x12 h1d y35 ffa fs15 fc0 sc0 ls2b">dx</div><div class="t me x7 h1d y36 ffa fs15 fc0 sc0 ls2d">d<span class="ffc ls1 ws0 v5"> <span class="_5e blank"></span>'<span class="_57 blank"></span>.<span class="_5f blank"></span> <span class="_60 blank"></span><span class="ls31 ws28">'.<span class="_61 blank"></span><span class="ls1 ws0"> <span class="_40 blank"></span>)<span class="_60 blank"></span>.<span class="_62 blank"></span>(<span class="_63 blank"> </span><span class="ffb fs16 ws2a v6">\uf0f2 \uf0f2<span class="_64 blank"></span>\uf0f2<span class="_65 blank"> </span><span class="fs15 ws1e v7">\uf02b<span class="_66 blank"></span>\uf03d</span></span></span></span></span></div></div><div class="c x32 y44 w9 h1b"><div class="t mf x34 h1e y39 ffb fs16 fc0 sc0 ls1 ws2b">\uf0f2<span class="_67 blank"></span>\uf0f2<span class="_68 blank"></span>\uf0f2 <span class="fs15 ws2c v7">\uf03d<span class="_66 blank"></span>\uf02d <span class="ffa ls2b ws27">dx<span class="_69 blank"></span><span class="ls1 ws1b">v<span class="_5d blank"></span>u<span class="_5c blank"></span><span class="ls2b ws27">dx<span class="_27 blank"></span><span class="ls1 ws1b">v<span class="_54 blank"></span>u<span class="_5b blank"></span><span class="ls2b ws27">dx<span class="_3d blank"></span><span class="ls1 ws1b">v<span class="_5d blank"></span>u</span></span></span></span></span></span></span></div><div class="t mf x12 h1d y35 ffa fs15 fc0 sc0 ls2b">dx</div><div class="t mf x7 h1d y36 ffa fs15 fc0 sc0 ls2e">d<span class="ffc ls1 ws0 v5">'<span class="_57 blank"></span>.<span class="_6a blank"></span> <span class="_60 blank"></span><span class="ls31 ws28">'.<span class="_6b blank"></span><span class="ls1 ws0"> <span class="_40 blank"></span>)<span class="_60 blank"></span>.<span class="_62 blank"></span>(</span></span></span></div></div><div class="t m0 x35 h2 y45 ff1 fs1 fc0 sc0 ls1 ws0"> </div><div class="t m0 x2c hc y1a ff1 fs8 fc2 sc0 ls1 ws0">Integr<span class="_3 blank"></span>al <span class="_3 blank"></span>Indefinida </div><div class="t m0 x35 h2 y46 ff1 fs1 fc0 sc0 ls1 ws0"> </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf6" class="pf w0 h0" data-page-no="6"><div class="pc pc6 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w0 h0" alt="" src="https://files.passeidireto.com/901d6444-6f43-4885-b13e-258399dbb0ec/bg6.png"><div class="t m0 x36 hb y47 ff4 fs5 fc4 sc0 ls4">\uf06e<span class="ff5 fs7 fc0 ls1 ws0">Rearranjando <span class="_4 blank"></span><span class="lse ws7">os</span> termos, temos:<span class="_4 blank"></span> </span></div><div class="t m0 x22 h7 y48 ff5 fs4 fc0 sc0 ls1 ws0"> </div><div class="t m0 x22 h7 y49 ff5 fs4 fc0 sc0 ls1 ws0"> </div><div class="t m0 x36 h7 y4a ff5 fs4 fc0 sc0 ls1 ws0"> <span class="_4 blank"></span> <span class="_5 blank"> </span><span class="ls6 ws3">que</span> é a fórmula <span class="_4 blank"> </span><span class="ls6 ws3">da</span> in<span class="_4 blank"></span>tegração<span class="_4 blank"></span> <span class="ls6 ws3">por</span> partes<span class="_4 blank"> </span>. </div><div class="t m0 x36 h23 y4b ff5 fs1a fc0 sc0 ls1 ws0"> </div><div class="t m0 x36 h7 y4c ff4 fs5 fc4 sc0 ls4">\uf06e<span class="ff5 fs4 fc0 ls1 ws0">Essa <span class="_6c blank"> </span>fórmula <span class="_6c blank"> </span>é <span class="_6c blank"> </span>mais <span class="_6c blank"> </span>facilmente<span class="_4 blank"> </span> <span class="_6c blank"> </span>lembrada <span class="_6c blank"> </span><span class="ls35 ws2d">na</span> <span class="_6c blank"> </span>forma <span class="_6c blank"> </span>diferencial<span class="_4 blank"></span>. </span></div><div class="t m0 x22 h7 y4d ff5 fs4 fc0 sc0 ls1 ws2e">Sejam:<span class="ffd ws0"> </span></div><div class="t m0 x20 h7 y4e ff4 fs2 fc3 sc0 ls32">\uf06e<span class="ffd fs4 fc0 ls1 ws0">u = f(x) <span class="_4 blank"> </span> <span class="ls6 ws3">du</span><span class="lsc"> = </span><span class="ffe ws2f">f\u2019(x)dx<span class="ff5 lsc">;</span></span> </span></div><div class="t m0 x20 hb y4f ff4 fs1b fc3 sc0 ls33">\uf06e<span class="ffd fs7 fc0 ls1 ws0">v = g(x) <span class="_4 blank"></span><span class="lse ws7">dv</span><span class="ls34"> = </span><span class="ffe ws30">g\u2019(x)dx</span><span class="ff5">. </span></span></div><div class="t m0 x36 h7 y50 ff4 fs5 fc4 sc0 ls4">\uf06e<span class="ff5 fs4 fc0 ls1 ws0">Usando <span class="_6d blank"> </span>a <span class="_6d blank"> </span>regra <span class="_6d blank"> </span><span class="ls6 ws3">de</span> <span class="_6d blank"> </span>substituição,<span class="_4 blank"> </span> <span class="_6d blank"> </span>a <span class="_6d blank"> </span>fórmula<span class="_4 blank"></span> <span class="_6d blank"> </span>acima <span class="_6d blank"> </span><span class="ls6 ws3">pode</span> <span class="_6e blank"> </span>ser </span></div><div class="t m0 x22 h7 y51 ff5 fs4 fc0 sc0 ls1 ws0">simplificada<span class="_4 blank"> </span> para:<span class="fc5"> </span></div><div class="c x37 y52 wa h24"><div class="t m10 x38 h25 y53 ffa fs1c fc0 sc0 ls36 ws31">dx<span class="_6f blank"></span><span class="ls1 ws32">x<span class="_70 blank"></span>g<span class="_c blank"></span>x<span class="_71 blank"></span>f<span class="_72 blank"></span>x<span class="_70 blank"></span>g<span class="_c blank"></span>x<span class="_22 blank"></span>f<span class="_73 blank"></span><span class="ls36 ws31">dx<span class="_1d blank"></span><span class="ls1 ws33">x<span class="_6f blank"></span>g<span class="_c blank"></span>x<span class="_51 blank"></span>f <span class="ffb fs1d ws34 v9">\uf0f2<span class="_74 blank"></span>\uf0f2 <span class="fs1c ws35 va">\uf02d<span class="_75 blank"></span>\uf03d <span class="ffc ws32">)<span class="_76 blank"></span>(<span class="_32 blank"></span><span class="ls37 ws36">).<span class="_32 blank"></span><span class="ls1 ws32">(<span class="_77 blank"></span>'<span class="_78 blank"></span>)<span class="_76 blank"></span>(<span class="_32 blank"></span><span class="ls37 ws36">).<span class="_32 blank"></span><span class="ls1 ws32">(<span class="_4a blank"></span>)<span class="_76 blank"></span>(<span class="_77 blank"></span>'<span class="_31 blank"></span><span class="ls37 ws36">).<span class="_32 blank"></span><span class="ls1">(</span></span></span></span></span></span></span></span></span></span></span></span></div></div><div class="t m0 x39 h2 y54 ff1 fs1 fc0 sc0 ls1 ws0"> </div><div class="t m0 x3a h1 y55 ff1 fs0 fc0 sc0 ls1 ws0"> </div><div class="c x3b y56 wb h26"><div class="t m11 x3c h27 y57 ffb fs1e fc0 sc0 ls1 ws37">\uf0f2<span class="_79 blank"></span>\uf0f2 <span class="fs1f ws38 vb">\uf02d<span class="_7a blank"></span>\uf03d <span class="ffa ws39">v du<span class="_7b blank"></span><span class="ls38 ws3a">uv<span class="_7c blank"></span><span class="ls1 ws3b">ud<span class="_2 blank"></span>v</span></span></span></span></div></div><div class="t m0 x1f hc y1a ff1 fs8 fc2 sc0 ls1 ws0">Integr<span class="_3 blank"></span>al <span class="_3 blank"></span>Indefinida </div><div class="c x37 y58 wc h28"><div class="t m12 x3d h29 y59 ffc fs20 fc0 sc0 ls1 ws0">,<span class="_a blank"></span> <span class="_62 blank"></span><span class="ls39 ws3c">'.<span class="_33 blank"></span><span class="ls1 ws3d">.<span class="_10 blank"></span>'<span class="_7d blank"></span>. <span class="ffa ls3a ws3e">dx<span class="_1f blank"></span><span class="ls1 ws3f">v<span class="_7e blank"></span>u<span class="_73 blank"></span>v<span class="_35 blank"></span>u<span class="_56 blank"></span><span class="ls3a ws3e">dx<span class="_13 blank"></span><span class="ls1 ws40">v<span class="_35 blank"></span>u <span class="ffb fs21 ws41 vc">\uf0f2<span class="_7f blank"></span>\uf0f2 <span class="fs20 ws42 v1">\uf02d<span class="_1a blank"></span>\uf03d</span></span></span></span></span></span></span></span></div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf7" class="pf w0 h0" data-page-no="7"><div class="pc pc7 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w0 h0" alt="" src="https://files.passeidireto.com/901d6444-6f43-4885-b13e-258399dbb0ec/bg7.png"><div class="t m0 x7 h2 y5a ff4 fs9 fc3 sc0 ls3b">\uf06e<span class="ff1 fs1 fc0 ls1 ws0"> <span class="ff2 fs22 fc5">Exempl<span class="_2 blank"></span>o 1: </span></span></div><div class="t m0 x3e h2a y5b ff4 fs2 fc4 sc0 ls3c">\uf06e<span class="ff1 fs22 fc0 ls1 ws0">Usando <span class="_2 blank"></span>o método da <span class="_2 blank"></span>integração por partes, <span class="_2 blank"></span>determine:<span class="fc5"> </span></span></div><div class="t m0 x7 h2 y5c ff4 fs9 fc3 sc0 ls3b">\uf06e<span class="ff1 fs1 fc5 ls1 ws0"> Solução </span></div><div class="t m0 x3e h2a y5d ff4 fs2 fc4 sc0 ls3c">\uf06e<span class="ff1 fs22 fc0 ls1 ws0">Usamos <span class="_2 blank"></span>a fórmula simpl<span class="_2 blank"></span>ificada da integração por p<span class="_2 blank"></span>artes, fazendo: </span></div><div class="t m0 x3f h2b y5e ff4 fs23 fc3 sc0 ls1">\uf06e</div><div class="t m13 x40 h2c y5e ff1 fs24 fc0 sc0 ls1 ws0">u = x,<span class="_2 blank"></span> du = dx; </div><div class="t m0 x3f h2b y5f ff4 fs23 fc3 sc0 ls1">\uf06e</div><div class="t m13 x40 h2c y5f ff1 fs24 fc0 sc0 ls1 ws0">v = senx, dv = cosxdx. </div><div class="t m0 x3e h2a y60 ff4 fs2 fc4 sc0 ls3c">\uf06e<span class="ff1 fs22 fc0 ls1 ws43">Então:<span class="fc5 ws0"> </span></span></div><div class="c x41 y61 wd h2d"><div class="t m14 x1d h2e y62 ffb fs25 fc0 sc0 ls3d">\uf0f2<span class="ffa fs26 ls1 ws44 vd">x<span class="_2 blank"></span>dx<span class="_80 blank"></span>x <span class="ffc ws45">c os<span class="_69 blank"></span>.</span></span></div></div><div class="c x16 y63 we h2f"><div class="t m15 x42 h30 y64 ffb fs27 fc0 sc0 ls1 ws46">\uf0f2<span class="_81 blank"></span>\uf0f2 <span class="fs28 ws47 va">\uf02d<span class="_82 blank"></span>\uf03d <span class="ffa ws48">v du<span class="_83 blank"></span><span class="ls3f ws49">uv<span class="_4a blank"></span><span class="ls1 ws4a">ud<span class="_2 blank"></span>v</span></span></span></span></div></div><div class="c x16 y65 wf h31"><div class="t m16 x43 h32 y66 ffb fs29 fc0 sc0 ls1 ws4b">\uf0f2<span class="_84 blank"></span>\uf0f2 <span class="fs2a ws4c ve">\uf02d<span class="_85 blank"></span>\uf03d <span class="ffa ls40 ws4d">dx<span class="_1a blank"></span><span class="ls1 ws4e">se<span class="_86 blank"></span>n<span class="_3 blank"></span>x<span class="_87 blank"></span>se<span class="_86 blank"></span>n<span class="_3 blank"></span>x<span class="_39 blank"></span>x<span class="_26 blank"></span><span class="ls40 ws4d">dx<span class="_a blank"></span><span class="ls1 ws4f">x<span class="_82 blank"></span>x <span class="ffc ws0"> <span class="_88 blank"></span>.<span class="_89 blank"></span> <span class="_8a blank"></span>c<span class="_86 blank"></span>o<span class="_3 blank"></span>s<span class="_8b blank"></span>.</span></span></span></span></span></span></div></div><div class="c x16 y67 w10 h33"><div class="t m17 x44 h34 y68 ffa fs2b fc0 sc0 ls1 ws50">c<span class="_15 blank"></span>x<span class="_67 blank"></span>se<span class="_86 blank"></span>n<span class="_3 blank"></span>x<span class="_8c blank"></span>x<span class="_8d blank"></span><span class="ls41 ws51">dx<span class="_8b blank"></span><span class="ls1 ws52">x<span class="_73 blank"></span>x <span class="ffb ws53">\uf02b<span class="_8e blank"></span>\uf02b<span class="_3a blank"></span>\uf03d</span></span></span></div><div class="t m17 x1d h35 y69 ffb fs2c fc0 sc0 ls3e">\uf0f2<span class="ffc fs2b ls1 ws0 ve">c<span class="_8f blank"></span>o<span class="_3 blank"></span>s<span class="_90 blank"></span>.<span class="_e blank"></span> <span class="_8a blank"></span>c<span class="_86 blank"></span>o<span class="_3 blank"></span>s<span class="_70 blank"></span>.</span></div></div><div class="t m0 x1f hc y1a ff1 fs8 fc2 sc0 ls1 ws0">Integr<span class="_3 blank"></span>al <span class="_3 blank"></span>Indefinida </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf8" class="pf w0 h0" data-page-no="8"><div class="pc pc8 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w0 h0" alt="" src="https://files.passeidireto.com/901d6444-6f43-4885-b13e-258399dbb0ec/bg8.png"><div class="t m0 x12 h2 y6a ff4 fs9 fc3 sc0 ls3b">\uf06e<span class="ff1 fs1 fc0 ls1 ws0"> <span class="fc5">Observações </span></span></div><div class="t m0 x13 h2 y6b ff4 fsa fc4 sc0 ls15">\uf06e<span class="ff1 fs1 fc0 ls1 ws0">O <span class="_91 blank"> </span>objetivo <span class="_91 blank"> </span><span class="ls43 ws54">da</span> <span class="_91 blank"> </span>integração <span class="_91 blank"> </span>por <span class="_91 blank"> </span>partes <span class="_91 blank"> </span>é <span class="_91 blank"> </span>passar <span class="_91 blank"> </span><span class="ls43 ws54">de</span> <span class="_91 blank"> </span>uma </span></div><div class="t m0 x16 h2 y6c ff1 fs1 fc0 sc0 ls1 ws55">integral<span class="ls42 ws0"> <span class="ls1">que <span class="_5 blank"> </span>não <span class="_7 blank"> </span>sabemos <span class="_5 blank"> </span>como <span class="_5 blank"> </span>calcular <span class="_7 blank"> </span>para <span class="_4 blank"> </span>uma </span></span></div><div class="t m0 x16 h2 y6d ff1 fs1 fc0 sc0 ls1 ws0">integral <span class="ls43 ws54">que<span class="_2 blank"></span><span class="ls1 ws0"> podemos calcular. </span></span></div><div class="t m0 x13 hd y6e ff4 fs2d fc4 sc0 ls44">\uf06e<span class="ff1 fs0 fc0 ls1 ws0">Geralmente, <span class="_8 blank"> </span>esc<span class="_2 blank"></span>olhemos <span class="_8 blank"> </span><span class="ff7 fsb ls43">dv</span></span></div><div class="t m13 x45 h36 y6e ff1 fs2e fc0 sc0 ls1 ws0"> </div><div class="t m0 x46 h1 y6e ff1 fs0 fc0 sc0 ls1 ws0">primeiro <span class="_92 blank"> </span>sendo <span class="_92 blank"> </span>a <span class="_92 blank"> </span>parte <span class="_92 blank"> </span><span class="ls49 ws56">do</span> </div><div class="t m0 x16 h37 y6f ff1 fs1 fc0 sc0 ls1 ws0">integrando, <span class="_93 blank"> </span>incluindo <span class="_93 blank"> </span><span class="ff7 fs3 lsb ws57">dx</span>, <span class="_93 blank"> </span><span class="ls43 ws54">que</span> <span class="_93 blank"> </span>sabemos <span class="_93 blank"> </span>int<span class="_4 blank"> </span>egrar <span class="_93 blank"> </span><span class="ls43 ws54">de</span> </div><div class="t m0 x16 h37 y70 ff1 fs1 fc0 sc0 ls1 ws0">maneira imediata; <span class="ff7 fs3 ls45">u</span> é a parte restante. </div><div class="t m0 x13 h2 y71 ff4 fsa fc4 sc0 ls15">\uf06e<span class="ff1 fs1 fc0 ls1 ws55">Lembre-<span class="ls4a ws58">se<span class="ls46 ws0"> <span class="ls43 ws54">de</span> <span class="ls43 ws54">que</span><span class="ls1"> <span class="_94 blank"> </span>a <span class="_94 blank"> </span>integração <span class="_94 blank"> </span><span class="ls43 ws54">por</span> <span class="_94 blank"> </span>partes <span class="_94 blank"> </span>nem <span class="_94 blank"> </span>sempre </span></span></span></span></div><div class="t m0 x16 h2 y72 ff1 fs1 fc0 sc0 ls1 ws55">funciona<span class="_4 blank"> </span>.<span class="fc5 ws0"> </span></div><div class="c x3a y73 w11 h38"><div class="t m18 x1d h39 y74 ffb fs2f fc0 sc0 ls47">\uf0f2<span class="ffa fs30 ls4b va">udv</span></div></div><div class="c x3a y75 w11 h38"><div class="t m18 x1d h39 y74 ffb fs2f fc0 sc0 ls48">\uf0f2<span class="ffa fs30 ls1 ws59 va">v d<span class="_4 blank"></span>u</span></div></div><div class="t m0 x11 hc y1a ff1 fs8 fc2 sc0 ls1 ws0">Integr<span class="_3 blank"></span>al <span class="_3 blank"></span>Indefinida </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf9" class="pf w0 h0" data-page-no="9"><div class="pc pc9 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w0 h0" alt="" src="https://files.passeidireto.com/901d6444-6f43-4885-b13e-258399dbb0ec/bg9.png"><div class="t m0 x1f hc y1a ff1 fs8 fc2 sc0 ls1 ws0">Integr<span class="_3 blank"></span>al <span class="_3 blank"></span>Indefinida </div><div class="t m0 x1 h3a y76 ff3 fs31 fc5 sc0 ls1 ws0">EXEMPLO 02 </div><div class="t m0 x1 h3b y77 ff5 fs31 fc0 sc0 ls1 ws0">Calcula<span class="_2 blank"></span>r </div><div class="c x47 y78 w12 h3c"><div class="t m19 x1d h3d y79 ffb fs32 fc0 sc0 ls4c">\uf0f2<span class="ffc fs33 ls4e ws5a v2">dx<span class="_c blank"></span><span class="ls1 ws5b">e<span class="_95 blank"></span>x <span class="fs34 vf">x</span></span></span></div></div><div class="t m0 x48 h3a y7a ff3 fs31 fc0 sc0 ls1 ws0">Solução </div><div class="t m0 x49 h3b y7b ff5 fs31 fc0 sc0 ls1 ws0">A<span class="_3 blank"></span> <span class="_2 blank"></span>integral dada deve ser escrita na forma <span class="_2 blank"></span> .<span class="_4 blank"> </span> </div><div class="c x4a y7c w13 h3c"><div class="t m1a x1d h3e y79 ffb fs32 fc0 sc0 ls4d">\uf0f2<span class="ffc fs33 ls4e ws5a v2">dv<span class="_96 blank"></span><span class="ls1">u</span></span></div></div><div class="t m0 x1 h3b y7d ff5 fs31 fc0 sc0 ls1 ws0">Seja, porta<span class="_2 blank"></span>nto: </div><div class="c x4b y7e w14 h3f"><div class="t m1b x4c h40 y7f ffc fs35 fc0 sc0 ls4f ws5c">dx<span class="_97 blank"></span><span class="ls1 ws5d">e<span class="_98 blank"></span>x <span class="fs36 v10">x</span></span></div><div class="t m1b x4d h41 y80 ffb fs37 fc0 sc0 ls1">\uf0f2</div></div><div class="c xf y81 w15 h42"><div class="t m1c x4e h43 y82 ffc fs38 fc0 sc0 ls1 ws5e">x<span class="_29 blank"></span>u <span class="ffb">\uf03d</span></div></div><div class="c x4f y83 w16 h44"><div class="t m1d x2 h45 y84 ffc fs39 fc0 sc0 ls50 ws5f">dx<span class="_51 blank"></span><span class="ls1 ws60">e<span class="_25 blank"></span><span class="ls50 ws61">dv <span class="fs3a ls1 vf">x</span></span></span></div><div class="t m1d x7 h46 y84 ffb fs39 fc0 sc0 ls1">\uf03d</div></div><div class="t m1 x50 h2 y85 ff1 fs1 fc0 sc0 ls1 ws0"> </div><div class="t m0 x51 h1 y86 ff1 fs0 fc0 sc0 ls1 ws0"> </div><div class="t m1 x51 h2 y87 ff1 fs1 fc0 sc0 ls1 ws0"> </div><div class="t m0 x26 h3b y88 ff5 fs31 fc0 sc0 ls1 ws0">Deste mod<span class="_2 blank"></span>o: </div><div class="c x1 y89 w17 h47"><div class="t m1e x52 h48 y8a ffc fs3b fc0 sc0 ls1 ws62">C<span class="_99 blank"></span>e<span class="_9a blank"></span><span class="ls51 ws63">xe<span class="_9b blank"></span>dx<span class="_c blank"></span><span class="ls1 ws62">e<span class="_9c blank"></span><span class="ls51 ws63">xe<span class="_9b blank"></span>du<span class="_9d blank"></span><span class="ls1 ws62">v<span class="_8d blank"></span><span class="ls51 ws63">uv<span class="_9e blank"></span>dv<span class="_54 blank"></span><span class="ls1 ws62">u<span class="_9f blank"></span><span class="ls51 ws64">dx<span class="_1d blank"></span>xe <span class="fs3c ls1 ws65 vd">x<span class="_a0 blank"></span>x<span class="_a1 blank"></span>x<span class="_a2 blank"></span>x<span class="_a3 blank"></span>x <span class="ffb fs3b ws66 v11">\uf02b<span class="_1d blank"></span>\uf02d<span class="_26 blank"></span>\uf03d<span class="_8e blank"></span>\uf02d<span class="_26 blank"></span>\uf03d<span class="_a4 blank"></span>\uf02d<span class="_a5 blank"></span>\uf03d<span class="_5c blank"></span>\uf03d <span class="fs3d ws67 v12">\uf0f2<span class="_a6 blank"></span>\uf0f2<span class="_7c blank"></span>\uf0f2<span class="_a7 blank"></span>\uf0f2</span></span></span></span></span></span></span></span></span></span></div></div><div class="t m0 x53 h49 y8b ff5 fs6 fc7 sc0 ls1 ws0">a constante C pode <span class="_4 blank"> </span>ser </div><div class="t m0 x53 h49 y8c ff5 fs6 fc7 sc0 ls1 ws0">incluída apenas no final. </div><div class="t m0 x2b h4a y8d ff5 fs22 fc8 sc0 ls1 ws0">INTEGRAÇÃO <span class="_2 blank"></span>POR P<span class="_86 blank"></span>AR<span class="_2 blank"></span>TES </div><div class="c x35 y8e w18 h4b"><div class="t m1f x54 h4c y8f ffc fs3e fc0 sc0 ls52 ws68">dx<span class="_9b blank"></span>du <span class="ffb ls1">\uf03d</span></div></div><div class="c x55 y90 w19 h47"><div class="t m20 x56 h4d y91 ffc fs3c fc0 sc0 ls1 ws69">x<span class="_a8 blank"></span>x<span class="_a9 blank"></span>x <span class="fs3b ws62 v11">e<span class="_aa blank"></span><span class="ls51 ws63">dx<span class="_8b blank"></span><span class="ls1 ws62">e<span class="_16 blank"></span>v<span class="_6a blank"></span><span class="ls51 ws63">dx<span class="_8b blank"></span><span class="ls1 ws62">e<span class="_4d blank"></span><span class="ls51 ws6a">dv <span class="ffb ls1 ws6b">\uf03d<span class="_33 blank"></span>\uf03d<span class="_73 blank"></span>\uf0ae<span class="_83 blank"></span>\uf03d <span class="fs3d ws67 v12">\uf0f2<span class="_ab blank"></span>\uf0f2<span class="_9b blank"></span>\uf0f2</span></span></span></span></span></span></span></span></div></div><div class="t m0 x57 h3b y92 ff5 fs31 fc0 sc0 ls1 ws0">Então: </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pfa" class="pf w0 h0" data-page-no="a"><div class="pc pca w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w0 h0" alt="" src="https://files.passeidireto.com/901d6444-6f43-4885-b13e-258399dbb0ec/bga.png"><div class="t m0 x1f hc y1a ff1 fs8 fc2 sc0 ls1 ws0">Integr<span class="_3 blank"></span>al <span class="_3 blank"></span>Indefinida </div><div class="t m0 x28 h3a y93 ff3 fs31 fc5 sc0 ls1 ws0">EXEMPLO 03 </div><div class="t m0 x28 h3b y94 ff5 fs31 fc0 sc0 ls1 ws0">Calcula<span class="_2 blank"></span>r </div><div class="c x58 y95 w1a h3c"><div class="t m21 x1d h3d y79 ffb fs32 fc0 sc0 ls53">\uf0f2<span class="fs34 ls54 v13">\uf02d</span><span class="ffc fs33 ls55 ws6c v2">dx<span class="_12 blank"></span><span class="ls1 ws6d">e<span class="_ac blank"></span>x <span class="fs34 ws6e vf">x<span class="_96 blank"></span>2</span></span></span></div></div><div class="t m0 x59 h3a y96 ff3 fs31 fc0 sc0 ls1 ws0">Solução </div><div class="t m0 x3e h3b y97 ff5 fs31 fc0 sc0 ls1 ws0">Seja: </div><div class="c x25 y98 w1b h4e"><div class="t m22 x5a h4f y99 ffc fs3f fc0 sc0 ls1">2</div><div class="t m22 x5b h50 y9a ffc fs40 fc0 sc0 ls1 ws6f">x<span class="_1f blank"></span>u <span class="ffb">\uf03d</span></div></div><div class="c x5c y98 w1c h51"><div class="t m23 x25 h52 y9b ffc fs41 fc0 sc0 ls56 ws70">dx<span class="_f blank"></span><span class="ls1 ws71">e<span class="_12 blank"></span><span class="ls56 ws72">dv <span class="fs42 ls1 ws73 v14">x<span class="_ad blank"></span><span class="ffb">\uf02d</span></span></span></span></div><div class="t m23 x7 h53 y9b ffb fs41 fc0 sc0 ls1">\uf03d</div></div><div class="t m0 xf h3b y9c ff5 fs31 fc0 sc0 ls1 ws0">Assim: </div><div class="c x25 y9d w1d h54"><div class="t m24 x5d h55 y9e ffc fs43 fc0 sc0 ls1 ws74">dx<span class="_a0 blank"></span>2x<span class="_58 blank"></span>du <span class="ffb">\uf03d</span></div></div><div class="c x25 y9f w1e h47"><div class="t m25 x5e h4d y91 ffc fs3c fc0 sc0 ls1 ws75">x<span class="_ae blank"></span>x<span class="_af blank"></span>x <span class="fs3b ws62 v11">e<span class="_b0 blank"></span><span class="ls51 ws63">dx<span class="_25 blank"></span><span class="ls1 ws62">e<span class="_16 blank"></span>v<span class="_6a blank"></span><span class="ls51 ws63">dx<span class="_25 blank"></span><span class="ls1 ws62">e<span class="_4d blank"></span><span class="ls51 ws76">dv <span class="ffb fs3c ls1 ws77 v14">\uf02d<span class="_b1 blank"></span>\uf02d<span class="_b2 blank"></span>\uf02d <span class="fs3b ws78 v11">\uf02d<span class="_62 blank"></span>\uf03d<span class="_b3 blank"></span>\uf03d<span class="_82 blank"></span>\uf0ae<span class="_b4 blank"></span>\uf03d <span class="fs3d ws67 v12">\uf0f2<span class="_b5 blank"></span>\uf0f2<span class="_9b blank"></span>\uf0f2</span></span></span></span></span></span></span></span></span></div></div><div class="t m0 xf h56 ya0 ff5 fs44 fc0 sc0 ls1 ws0">Portanto: </div><div class="c x3e ya1 w1f h57"><div class="t m26 x5f h58 ya2 ffc fs45 fc0 sc0 ls1 ws79">2x<span class="_2 blank"></span>dx<span class="_8a blank"></span>)<span class="_2f blank"></span>e<span class="_4e blank"></span>(<span class="_b6 blank"></span>e<span class="_ac blank"></span>x<span class="_9f blank"></span><span class="ls57 ws7a">du<span class="_20 blank"></span><span class="ls1 ws79">v<span class="_9b blank"></span><span class="ls57 ws7a">uv<span class="_9b blank"></span>dv<span class="_37 blank"></span><span class="ls1 ws79">u<span class="_b6 blank"></span><span class="ls57 ws7a">dx<span class="_25 blank"></span><span class="ls1 ws7b">e<span class="_b7 blank"></span>x <span class="fs46 ws7c v14">x<span class="_33 blank"></span>x<span class="_76 blank"></span>2<span class="_b8 blank"></span>x<span class="_76 blank"></span>2 <span class="ffb fs47 ws7d v15">\uf0f2<span class="_b9 blank"></span>\uf0f2<span class="_ba blank"></span>\uf0f2<span class="_bb blank"></span>\uf0f2 <span class="fs46 ws7e v16">\uf02d<span class="_1c blank"></span>\uf02d<span class="_bc blank"></span>\uf02d <span class="fs45 ws7f v11">\uf02d<span class="_71 blank"></span>\uf02d<span class="_72 blank"></span>\uf02d<span class="_bd blank"></span>\uf03d<span class="_3f blank"></span>\uf02d<span class="_be blank"></span>\uf03d<span class="_97 blank"></span>\uf03d</span></span></span></span></span></span></span></span></span></span></div></div><div class="t m0 x60 h4a ya3 ff5 fs22 fc8 sc0 ls1 ws0">INTEGRAÇÃO <span class="_2 blank"></span>POR P<span class="_86 blank"></span>AR<span class="_2 blank"></span>TES </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div>
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