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UNIVERSIDADE FEDERAL DO ABC Tabela de Derivadas, Integrais e Identidades Trigonome´tricas Derivadas Regras de Derivac¸a˜o • (cf(x)) 0 = cf 0(x) • Derivada da Soma (f(x) + g(x)) 0 = f 0(x) + g 0(x) • Derivada do Produto (f(x)g(x)) 0 = f 0(x)g(x) + f(x)g 0(x) • Derivada do Quociente✓ f(x) g(x) ◆ 0 = f 0(x)g(x)- f(x)g 0(x) g(x)2 • Regra da Cadeia (f(g(x)) 0 = (f 0(g(x))g 0(x) Func¸o˜es Simples • ddxc = 0 • ddxx = 1 • ddxcx = c • ddxxc = cxc-1 • ddx � 1 x � = ddx � x-1 � = -x-2 = - 1 x2 • ddx � 1 xc � = ddx (x -c) = - c xc+1 • ddx p x = ddxx 1 2 = 12x - 1 2 = 1 2 p x , Func¸o˜es Exponenciais e Logar´ıtmicas • ddxex = ex • ddx ln(x) = 1x • ddxax = ax ln(a) Func¸o˜es Trigonome´tricas • ddx sen x = cos x • ddx cos x = -sen x, • ddx tg x = sec2 x • ddx sec x = tg x sec x • ddx cotg x = -cossec 2x • ddx cossec x = -cossec x cotg x Func¸o˜es Trigonome´tricas Inversas • ddx arcsen x = 1p1-x2 • ddx arccos x = -1p1-x2 • ddx arctg x = 11+x2 • ddx arcsec x = 1|x|px2-1 • ddx arccotg x = -11+x2 • ddx arccossec x = -1|x|px2-1 Func¸o˜es Hiperbo´licas • ddx senh x = cosh x = e x+e-x 2 • ddx cosh x = senh x = e x-e-x 2 • ddx tgh x = sech2 x • ddx sech x = - tgh x sech x • ddx cotgh x = - cossech2 x Func¸o˜es Hiperbo´licas Inversas • ddx csch x = - coth x cossech x • ddx arcsenh x = 1px2+1 • ddx arccosh x = 1px2-1 • ddx arctgh x = 11-x2 • ddx arcsech x = -1xp1-x2 • ddx arccoth x = 11-x2 • ddx arccossech x = -1|x|p1+x2 1 Integrais Regras de Integrac¸a˜o • R cf(x)dx = c R f(x)dx • R[f(x) + g(x)]dx = R f(x)dx+ R g(x)dx • R f 0(x)g(x)dx = f(x)g(x)- R f(x)g 0(x)dx Func¸o˜es Racionais • R xn dx = xn+1n+1 + c para n 6= -1 • Z 1 x dx = ln |x| + c • Z du 1+ u2 = arctgu+ c • Z 1 a2 + x2 dx = 1 a arctg(x/a) + c • Z du 1- u2 = � arctgh u+ c, se |u| < 1 arccotgh u+ c, se |u| > 1 = 1 2 ln �� 1+u 1-u ��+ c Func¸o˜es Logar´ıtmicas • R ln xdx = x ln x- x+ c • R loga x dx = x loga x- xlna + c Func¸o˜es Irracionais • Z dup 1- u2 = arcsenu+ c • Z du u p u2 - 1 = arcsec u+ c • Z dup 1+ u2 = arcsenh u+ c = ln |u+ p u2 + 1| + c • Z dup 1- u2 = arccosh u+ c = ln |u+ p u2 - 1| + c • Z du u p 1- u2 = -arcsech |u| + c • Z du u p 1+ u2 = -arccosech |u| + c • Z 1p a2 - x2 dx = arcsen x a + c • Z -1p a2 - x2 dx = arccos x a + c Func¸o˜es Trigonome´tricas • R cos xdx = sen x+ c • R sen xdx = - cos x+ c • R tg xdx = ln |sec x| + c • R csc xdx = ln |csc x- cot x| + c • R sec xdx = ln |sec x+ tg x| + c • R cot xdx = ln |sen x| + c • R sec x tg xdx = sec x+ c • R csc x cot xdx = - csc x+ c • R sec2 x dx = tg x+ c • R csc2 x dx = - cot x+ c • R sen2 x dx = 12(x- sen x cos x) + c • R cos2 x dx = 12(x+ sen x cos x) + c Func¸o˜es Hiperbo´licas • R sinh xdx = cosh x+ c • R cosh xdx = sinh x+ c • R tgh xdx = ln(cosh x) + c • R csch xdx = ln ��tgh x2 ��+ c • R sech xdx = arctg(sinh x) + c • R coth xdx = ln | sinh x| + c 2 Identidades Trigonome´tricas 1. sen(90o - ✓) = cos ✓ 2. cos(90o - ✓) = sen ✓ 3. sen ✓ cos ✓ = tg ✓ 4. sen2 ✓+ cos2 ✓ = 1 5. sec2 ✓- tg2 ✓ = 1 6. csc2 ✓- cot2 ✓ = 1 7. sen 2✓ = 2 sen ✓ cos ✓ 8. cos 2✓ = cos2 ✓- sen2 ✓ = 2 cos2 ✓- 1 9. sen 2✓ = 2 sen ✓ cos ✓ 10. sen(↵± �) = sen↵ cos�± cos↵ sen� 11. cos(↵± �) = cos↵ sen�± sen↵ cos� 12. tg(↵± �) = tg↵± tg� 1⌥ tg↵ tg� 13. sen↵± sen� = 2 sen 1 2 (↵± �) cos 1 2 (↵± �) 14. cos↵+ cos� = 2 cos 1 2 (↵+ �) cos 1 2 (↵- �) 15. cos↵- cos� = 2 sen 1 2 (↵+ �) sen 1 2 (↵- �) 3
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