Calculo_de_integrais_indefinidos_com_aplicacao_das_propriedades

Prévia do material em texto

EXERCÍCIOS M1 Anatolie Sochirca ACM DEETC ISEL 
 
 1 
Cálculo de integrais indefinidos com aplicação das propriedades e das 
 
fórmulas da tabela dos integrais (integração imediata). 
 
 
 Para aplicar o método de integração imediata transformamos a expressão sob 
o sinal integral com o objectivo de obter um integral ou uma soma algébrica de 
integrais da tabela dos integrais. 
Neste caso é útil a transformação: se )(xG é uma primitiva evidente da função 
)(xg e ))(()( xGuxf = então 
( )∫∫∫ ⋅=⋅′⋅=⋅⋅ )())(()())(()()( xGdxGuxdxGxGuxdxgxf . 
 
 
►1) =⋅








−−+
⋅
−=⋅
−−+⋅−
∫∫ xd
xx
x
x
x
x
xx
x
x
xd
x
xxxxx
33
5 3
333
3
3
5 33 223223
 
 
=⋅







−−+−=⋅










−−+
⋅
−= ∫∫
−−−−+−
xdxxxxxxd
xx
x
x
x
x
xx
x
x 3
1
3
1
5
3
3
11
3
1
2
11
3
13
3
1
3
1
5
3
3
1
3
1
2
1
3
1
3
223223
 
=⋅







−−+−= ∫
−
xdxxxxx 3
1
15
4
3
2
6
7
3
8
223 
 
∫∫∫∫∫ =⋅−⋅−⋅+⋅−⋅=
−
xdxxdxxdxxdxxdx 3
1
15
4
3
2
6
7
3
8
223 
 
∫∫∫∫∫ =⋅⋅−⋅−⋅⋅+⋅⋅−⋅=
−
xdxxdxxdxxdxxdx 3
1
15
4
3
2
6
7
3
8
223 
 
=+
+−
⋅−
+
−
+
⋅+
+
⋅−
+
=
+−++++
Cxxxxx
1
3
121
15
41
3
221
6
7
3
1
3
8
1
3
11
15
41
3
21
6
71
3
8
 
 
=+⋅−−⋅+⋅−= Cxxxxx
3
22
15
19
3
52
6
133
3
11
3
2
15
19
3
5
6
13
3
11
 
 
=+⋅⋅−⋅−⋅⋅+⋅⋅−⋅= Cxxxxx 3
2
15
19
3
5
6
13
3
11
2
32
19
15
5
32
13
63
11
3
 
 
Cxxxxx +⋅−⋅−⋅+⋅−⋅= 3
2
15
19
3
5
6
13
3
11
3
19
15
5
6
13
18
11
3
. ■ 
EXERCÍCIOS M1 Anatolie Sochirca ACM DEETC ISEL 
 
 2 
►2) =⋅
+
=⋅=⋅=⋅
⋅
∫∫∫∫ )(5
1)()5(
1
)5(
1
)5(
1
xnld
xnlnl
xnld
xnlx
xd
xnl
xd
xnlx
 
 
( ) ( ) CxnlnlCxnlnlnlxnlnld
xnlnl
+=++=+⋅
+
= ∫ )5(5)5(5
1
. ■ 
 
Outro método: 
 
► =
⋅
⋅
⋅=
⋅
⋅
⋅=⋅
⋅⋅
=⋅
⋅
∫∫∫∫ x
xd
xnlx
xd
xnl
xd
xnlx
xd
xnlx 5
)5(
)5(
1
5
5
)5(
1
)5(5
5
)5(
1
 
 
( ) Cxnlnlxnld
xnl
+=⋅= ∫ )5()5(()5(
1
. ■ 
 
 
►3) ( )=⋅=
+
⋅=⋅
+ ∫∫∫
xarctgdxarctg
x
xd
xarctgxd
x
xarctg 3
2
3
2
3
11
 
 
( ) CxarctgCxarctg +⋅=+
+
=
+
4
13
4
1
13
. ■ 
 
 
►4) =
+++
⋅+⋅=⋅
++
+⋅
∫∫ 196
)3(2
106
)3(2
22 xx
xd
xarctgxd
xx
xarctg
 
 
( ) =+⋅+⋅=
++
+
⋅+⋅= ∫∫ )3()3(21)3(
)3()3(2 2 xarctgdxarctgx
xd
xarctg 
 
CxarctgCxarctg ++=+
+
+
⋅=
+
)3(
11
)3(2 2
11
. ■ 
 
 
►5) ( )
( )
( ) ( ) =+
⋅−⋅
=
+
⋅−
=⋅
+
−
∫∫∫ 222222
)2()2()2(
xsenxosc
xdxoscxdxsen
xsenxosc
xdxoscxsen
xd
xsenxosc
xoscxsen
 
 
( ) ( ) =+
−−⋅⋅
=
+
⋅−⋅⋅⋅
= ∫∫ 2222
)()(22
xsenxosc
senxdxoscdxosc
xsenxosc
xdxoscxdxoscxsen
 
 
( ) ( ) =+
+⋅⋅
−=
+
−⋅⋅−
= ∫∫ 2222
)()(2)()(2
xsenxosc
senxdxoscdxosc
xsenxosc
senxdxoscdxosc
 
 
( ) ( ) =+
+
−=
+
+
−= ∫∫ 22
2
22
2 )()()(
xsenxosc
senxxoscd
xsenxosc
senxdxoscd
 
 
EXERCÍCIOS M1 Anatolie Sochirca ACM DEETC ISEL 
 
 3 
( ) ( ) =+
+−
+
−=+⋅+−=
+−
−
∫ C
xsenxosc
senxxoscdxsenxosc
12
)(
122
222
 
 
( ) C
xsenxosc
Cxsenxosc +
+
=+
−
+
−=
−
2
12 1
1
. ■ 
 
►6) ( ) =
−
⋅
′
⋅=
−
⋅⋅
⋅=
−
⋅⋅⋅
=⋅
−
∫∫∫∫ 2
2
222 12
1
1
2
2
1
1
2
2
1
1 x
xdx
x
xdx
x
xdx
xd
x
x
 
 
( ) =−
−
⋅−=
−
−
⋅−=
−
−−
⋅=
−
⋅= ∫∫∫∫
2
1
2
2
2
2
2
2
2
2
1
)1(
2
1
1
)1(
2
1
1
)(
2
1
1
)(
2
1
x
xd
x
xd
x
xd
x
xd
 
 
( ) ( ) =+
+−
−
⋅−=−⋅−⋅−=
+−
−
∫ C
x
xdx
1
2
1
1
2
1)1(1
2
1 12
1
2
22
1
2
 
( ) CxCx +−−=+−−= 2212 11 . ■ 
 
►7) ( ) =
−
⋅
′
⋅=
−
⋅⋅
⋅=
−
⋅⋅⋅
=⋅
−
∫∫∫∫ 4
2
444 12
1
1
2
2
1
1
2
2
1
1 x
xdx
x
xdx
x
xdx
xd
x
x
 
 
( ) ( ) Cxarcsenx
xd
x
xd
+⋅=
−
⋅=
−
⋅= ∫∫ )(2
1
1
)(
2
1
1
)(
2
1 2
22
2
22
2
. ■ 
 
 
 
►8) =⋅






−
+
−
⋅
=⋅
−
+⋅
∫∫ xd
x
x
x
xarcsen
xd
x
xxarcsen
222 11
2
1
2
 
 
=⋅
−
+⋅
−
⋅⋅=⋅
−
+⋅
−
⋅
= ∫∫∫∫
4434421
)6
2222 11
12
11
2
exemplover
xd
x
x
xd
x
xarcsenxd
x
x
xd
x
xarcsen
 
 
( ) =+−−+⋅′⋅⋅= ∫ Cxxdxarcsenxarcsen 212 
 
CxxarcsenCxxarcsen +−−=+−−⋅⋅= 2222 11
2
12 . ■ 
 
 
EXERCÍCIOS M1 Anatolie Sochirca ACM DEETC ISEL 
 
 4 
►9) ( ) =
+⋅⋅
⋅+⋅⋅
=
+⋅⋅
⋅+⋅
=⋅
+⋅⋅
+⋅
∫∫∫
xxx
xdxdx
xxx
xdx
xd
xxx
x
2
13
2
13
2
13
 
 
=
+⋅
+







⋅
=
+⋅
+⋅
′








⋅⋅
=
+⋅⋅
+⋅⋅
= ∫∫∫
xx
xdxd
xx
xdxdx
xxx
xdxdx
2
3
2
3
2
3
2
3
2
1
2
1
2
2
2
3
23
2
3
 
 
Cxxnl
xx
xxd
xx
xdxd
++⋅=
+⋅








+⋅
=
+⋅
+







⋅
= ∫∫ 2
3
2
3
2
3
2
3
2
3
2
2
2
2
2
. ■ 
 
 
 
►10) ( ) ( ) ( ) ( ) ( ) =⋅+=⋅′⋅+=⋅⋅+ ∫∫∫ xxxxxx edexdeexdee pipipi 222 
 
( ) ( ) ( ) Ceede xxx +
+
+
=+⋅+=
+
∫ 1
222
1
pi
pi
pi
. ■ 
 
►11) ( ) ( )( ) ( ) =⋅⋅
⋅
=⋅
′
⋅⋅
⋅
=⋅⋅=⋅⋅ ∫∫∫∫
xxxxx ed
enl
xde
enl
xdexde 3)3(
13)3(
133 
 
( ) ( ) ( ) C
nl
eC
enlnl
eCe
enl
xx
x +
+
⋅
=+
+
⋅
=+⋅⋅
⋅
=
13
3
3
33)3(
1
. ■ 
 
 
►12) =⋅





−=⋅
−
=⋅=⋅ ∫∫∫∫ xdxosc
xosc
xosc
xd
xosc
xosc
xd
xosc
xsen
xdxtg 2
2
22
2
2
2
2 11
 
 
( ) =+−⋅′=⋅−⋅=⋅





−= ∫∫∫∫ Cxxdxtgxdxdxosc
xd
xosc
1111 22 
 
( ) CxxtgCxxtgd +−=+−= ∫ . ■ 
 
 
►13) Ceed
x
dexd
x
exd
x
e
xxxx
x
+−=







−=





⋅−=⋅
′






⋅−=⋅ ∫∫∫∫
1111
2
1
11
. ■ 
 
EXERCÍCIOS M1 Anatolie Sochirca ACM DEETC ISEL 
 
 5 
►14) ( ) ( ) =⋅⋅=⋅′⋅⋅=⋅⋅⋅⋅=⋅⋅ ∫∫∫∫ 222222 72
1
2
172
2
177 xdxdxxdxxdx xxxx 
 
( ) ( ) C
nl
d
nl
xdnl
nl
x
xx +
⋅
=⋅⋅=⋅⋅⋅⋅= ∫∫ 72
77
7
1
2
177
7
1
2
1
2
222
. ■ 
 
►15) =




 −
=⇔⋅−==⋅∫ 2
)2(121)2()6( 222 αααα oscsensenoscxdxsen 
 
=⋅−⋅=⋅





−=⋅
−
= ∫∫∫∫ xd
xosc
xdxdxoscxdxosc
2
)12(
2
1
2
)12(
2
1
2
)12(1
 
 
( ) =⋅′⋅⋅−⋅=⋅⋅−⋅= ∫∫∫∫ xdxsenxdxdxoscxd )12(12
1
2
1
2
1)12(
2
1
2
1
 
 
( ) Cxsenxxsendxd +−=⋅−⋅= ∫∫ 24
)12(
2
)12(
24
1
2
1
. ■ 
 
 
►16) =⋅
⋅
+
=⋅
⋅
=
⋅
∫∫∫ xdxoscxsen
xoscxsen
xd
xoscxsenxoscxsen
xd 221
 
 
=⋅





+=⋅




⋅
+
⋅
= ∫∫ xdxsen
xosc
xosc
xsen
xd
xoscxsen
xosc
xoscxsen
xsen 22
 
 
( ) ( )
=⋅
′
+⋅
′
−=⋅+⋅= ∫∫∫∫ xdxsen
xsen
xd
xosc
xosc
xd
xsen
xosc
xd
xosc
xsen
 
 
( ) ( ) ( ) ( ) =⋅′−⋅′=⋅′−⋅′= ∫∫∫∫ xdxoscnlxdxsennlxdxosc
xosc
xd
xsen
xsen
 
 
CxtgnlC
xosc
xsen
nlCxoscnlxsennl +=+=+−= . ■ 
 
 
►17) ( ) ( ) =
+
⋅
′+
−=
+
⋅
′
−=⋅
+ ∫∫∫ xosc
xdxosc
xosc
xdxosc
xd
xosc
senx
5
5
55
 
 
( ) Cxoscnlxdxoscnl ++−=⋅′+−= ∫ 55 . ■ 
 
 
 
EXERCÍCIOS M1 Anatolie Sochirca ACM DEETC ISEL 
 
 6 
►18) =⋅⋅=⋅
−
⋅⋅⋅
=⋅
−
⋅
∫∫∫ xd
xosc
xsen
xd
xsenxosc
xoscsenx
xd
xsenxosc
xoscsenx
)2(
)2(
2
122
1
2222
 
 
( )
( ) ( ) =⋅⋅−=⋅
′
⋅−
⋅= ∫∫
− )2()2(
4
1
)2(
)2(
2
1
2
1
2
1
xoscdxoscxd
xosc
xosc
 
 
( ) ( ) ( ) CxoscCxoscCxosc +⋅−=+⋅−=+
+−
⋅−=
+−
2
12
11
2
1
)2(
2
1
2
1
)2(
4
1
1
2
1
)2(
4
1
. ■ 
 
Outro método: 
 
► 
( ) ( )
=
⋅−
⋅−⋅−
⋅=
−
⋅
′
⋅⋅⋅
=⋅
−
⋅
∫∫∫
xsen
xsend
xsenxosc
xdsenxsenx
xd
xsenxosc
xoscsenx
2
2
2222 21
2
2
1
2
122
1
 
 ( ) ( ) ( ) =⋅−⋅⋅−⋅−=
⋅−
⋅−
⋅−= ∫∫
−
xsendxsen
xsen
xsend 22
1
2
2
2
2121
4
1
21
21
4
1
 
 
( ) ( ) ( ) CxoscCxsenCxsen +⋅−=+⋅−⋅=+
+−
⋅−
⋅−=
+−
2
1
2
1
2
1
2
1
2
)2(
2
121
2
1
1
2
1
21
4
1
. ■ 
 
 
►19) =⋅⋅⋅⋅⋅+=⋅⋅⋅+ ∫∫ xdxoscxsenxoscxdxsenxosc 254)2(54 22 
 
( ) ( ) =⋅′⋅⋅⋅⋅+−=⋅′−⋅⋅⋅⋅+= ∫∫ xdxoscxoscxoscxdxoscxoscxosc 254254 22 
 
( ) ( ) =⋅⋅⋅+⋅−=⋅⋅+−= ∫∫ xoscdxoscxoscdxosc 2222 5545
154 
 
( ) ( ) ( ) =+
+
⋅+
⋅−=⋅+⋅⋅+⋅−=
+
∫ C
xosc
xoscdxosc
1
2
1
54
5
15454
5
1 12
1
2
22
1
2
 
 
( ) Cxosc +⋅+⋅−= 23254
15
2
. ■