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UNIVERSIDADE FEDERAL DO ABC
Tabela de Derivadas, Integrais e Identidades Trigonom´etricas
Derivadas
Regras de Deriva¸c˜ao
• (cf(x)) = cf (x)
• Derivada da Soma
(f(x) + g(x)) = f (x) + g (x)
• Derivada do Produto
(f(x)g(x)) = f (x)g(x) + f(x)g (x)
• Derivada do Quociente
f(x)
g(x)
=
f (x)g(x) − f(x)g (x)
g(x)2
• Regra da Cadeia
(f(g(x)) = (f (g(x))g (x)
Fun¸c˜oes Simples
• d
dx c = 0
• d
dx x = 1
• d
dx cx = c
• d
dx xc = cxc−1
• d
dx
1
x = d
dx x−1 = −x−2 = − 1
x2
• d
dx
1
xc = d
dx (x−c) = − c
xc+1
• d
dx
√
x = d
dx x
1
2 = 1
2 x−1
2 = 1
2
√
x
,
Fun¸c˜oes Exponenciais e Logar´ıtmicas
• d
dx ex = ex
• d
dx ln(x) = 1
x
• d
dx ax = ax ln(a)
Fun¸c˜oes Trigonom´etricas
• d
dx sen x = cos x
• d
dx cos x = −sen x,
• d
dx tg x = sec2 x
• d
dx sec x = tg x sec x
• d
dx cotg x = −cossec 2x
• d
dx cossec x = −cossec x cotg x
Fun¸c˜oes Trigonom´etricas Inversas
• d
dx arcsen x = 1√
1−x2
• d
dx arccos x = −1√
1−x2
• d
dx arctg x = 1
1+x2
• d
dx arcsec x = 1
|x|
√
x2−1
• d
dx arccotg x = −1
1+x2
• d
dx arccossec x = −1
|x|
√
x2−1
Fun¸c˜oes Hiperb´olicas
• d
dx senh x = cosh x = ex+e−x
2
• d
dx cosh x = senh x = ex−e−x
2
• d
dx tgh x = sech2
x
• d
dx sech x = − tgh x sech x
• d
dx cotgh x = − cossech2
x
Fun¸c˜oes Hiperb´olicas Inversas
• d
dx csch x = − coth x cossech x
• d
dx arcsenh x = 1√
x2+1
• d
dx arccosh x = 1√
x2−1
• d
dx arctgh x = 1
1−x2
• d
dx arcsech x = −1
x
√
1−x2
• d
dx arccoth x = 1
1−x2
• d
dx arccossech x = −1
|x|
√
1+x2
1
Integrais
Regras de Integra¸c˜ao
• cf(x) dx = c f(x) dx
• [f(x) + g(x)] dx = f(x) dx + g(x) dx
• f (x)g(x) dx = f(x)g(x) − f(x)g (x) dx
Fun¸c˜oes Racionais
• xn dx = xn+1
n+1 + c para n = −1
•
1
x
dx = ln |x| + c
•
du
1 + u2
= arctg u + c
•
1
a2 + x2
dx =
1
a
arctg(x/a) + c
•
du
1 − u2
=
arctgh u + c, se |u| < 1
arccotgh u + c, se |u| > 1
=
1
2 ln 1+u
1−u + c
Fun¸c˜oes Logar´ıtmicas
• ln x dx = x ln x − x + c
• loga x dx = x loga x − x
ln a + c
Fun¸c˜oes Irracionais
•
du
√
1 − u2
= arcsen u + c
•
du
u
√
u2 − 1
= arcsec u + c
•
du
√
1 + u2
= arcsenh u + c
= ln |u +
√
u2 + 1| + c
•
du
√
1 − u2
= arccosh u + c
= ln |u +
√
u2 − 1| + c
•
du
u
√
1 − u2
= −arcsech |u| + c
•
du
u
√
1 + u2
= −arccosech |u| + c
•
1
√
a2 − x2
dx = arcsen
x
a
+ c
•
−1
√
a2 − x2
dx = arccos
x
a
+ c
Fun¸c˜oes Trigonom´etricas
• cos x dx = sen x + c
• sen x dx = − cos x + c
• tg x dx = ln |sec x| + c
• csc x dx = ln |csc x − cot x| + c
• sec x dx = ln |sec x + tg x| + c
• cot x dx = ln |sen x| + c
• sec x tg x dx = sec x + c
• csc x cot x dx = − csc x + c
• sec2 x dx = tg x + c
• csc2 x dx = − cot x + c
• sen2 x dx = 1
2 (x − sen x cos x) + c
• cos2 x dx = 1
2 (x + sen x cos x) + c
Fun¸c˜oes Hiperb´olicas
• sinh x dx = cosh x + c
• cosh x dx = sinh x + c
• tgh x dx = ln(cosh x) + c
• csch x dx = ln tgh x
2 + c
• sech x dx = arctg(sinh x) + c
• coth x dx = ln | sinh x| + c
2
Identidades Trigonom´etricas
1. sen(90o
− θ) = cos θ
2. cos(90o
− θ) = sen θ
3.
sen θ
cos θ
= tg θ
4. sen2
θ + cos2
θ = 1
5. sec2
θ − tg2
θ = 1
6. csc2
θ − cot2
θ = 1
7. sen 2θ = 2 sen θ cos θ
8. cos 2θ = cos2
θ − sen2
θ = 2 cos2
θ − 1
9. sen 2θ = 2 sen θ cos θ
10. sen(α ± β) = sen α cos β ± cos α sen β
11. cos(α ± β) = cos α cos β sen α sen β
12. tg(α ± β) =
tg α ± tg β
1 tg α tg β
13. sen α ± sen β = 2 sen
1
2
(α ± β) cos
1
2
(α ± β)
14. cos α + cos β = 2 cos
1
2
(α + β) cos
1
2
(α − β)
15. cos α − cos β = 2 sen
1
2
(α + β) sen
1
2
(α − β)
3

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Tabela completa de derivadas e integrais

  • 1. UNIVERSIDADE FEDERAL DO ABC Tabela de Derivadas, Integrais e Identidades Trigonom´etricas Derivadas Regras de Deriva¸c˜ao • (cf(x)) = cf (x) • Derivada da Soma (f(x) + g(x)) = f (x) + g (x) • Derivada do Produto (f(x)g(x)) = f (x)g(x) + f(x)g (x) • Derivada do Quociente f(x) g(x) = f (x)g(x) − f(x)g (x) g(x)2 • Regra da Cadeia (f(g(x)) = (f (g(x))g (x) Fun¸c˜oes Simples • d dx c = 0 • d dx x = 1 • d dx cx = c • d dx xc = cxc−1 • d dx 1 x = d dx x−1 = −x−2 = − 1 x2 • d dx 1 xc = d dx (x−c) = − c xc+1 • d dx √ x = d dx x 1 2 = 1 2 x−1 2 = 1 2 √ x , Fun¸c˜oes Exponenciais e Logar´ıtmicas • d dx ex = ex • d dx ln(x) = 1 x • d dx ax = ax ln(a) Fun¸c˜oes Trigonom´etricas • d dx sen x = cos x • d dx cos x = −sen x, • d dx tg x = sec2 x • d dx sec x = tg x sec x • d dx cotg x = −cossec 2x • d dx cossec x = −cossec x cotg x Fun¸c˜oes Trigonom´etricas Inversas • d dx arcsen x = 1√ 1−x2 • d dx arccos x = −1√ 1−x2 • d dx arctg x = 1 1+x2 • d dx arcsec x = 1 |x| √ x2−1 • d dx arccotg x = −1 1+x2 • d dx arccossec x = −1 |x| √ x2−1 Fun¸c˜oes Hiperb´olicas • d dx senh x = cosh x = ex+e−x 2 • d dx cosh x = senh x = ex−e−x 2 • d dx tgh x = sech2 x • d dx sech x = − tgh x sech x • d dx cotgh x = − cossech2 x Fun¸c˜oes Hiperb´olicas Inversas • d dx csch x = − coth x cossech x • d dx arcsenh x = 1√ x2+1 • d dx arccosh x = 1√ x2−1 • d dx arctgh x = 1 1−x2 • d dx arcsech x = −1 x √ 1−x2 • d dx arccoth x = 1 1−x2 • d dx arccossech x = −1 |x| √ 1+x2 1
  • 2. Integrais Regras de Integra¸c˜ao • cf(x) dx = c f(x) dx • [f(x) + g(x)] dx = f(x) dx + g(x) dx • f (x)g(x) dx = f(x)g(x) − f(x)g (x) dx Fun¸c˜oes Racionais • xn dx = xn+1 n+1 + c para n = −1 • 1 x dx = ln |x| + c • du 1 + u2 = arctg u + c • 1 a2 + x2 dx = 1 a arctg(x/a) + c • du 1 − u2 = arctgh u + c, se |u| < 1 arccotgh u + c, se |u| > 1 = 1 2 ln 1+u 1−u + c Fun¸c˜oes Logar´ıtmicas • ln x dx = x ln x − x + c • loga x dx = x loga x − x ln a + c Fun¸c˜oes Irracionais • du √ 1 − u2 = arcsen u + c • du u √ u2 − 1 = arcsec u + c • du √ 1 + u2 = arcsenh u + c = ln |u + √ u2 + 1| + c • du √ 1 − u2 = arccosh u + c = ln |u + √ u2 − 1| + c • du u √ 1 − u2 = −arcsech |u| + c • du u √ 1 + u2 = −arccosech |u| + c • 1 √ a2 − x2 dx = arcsen x a + c • −1 √ a2 − x2 dx = arccos x a + c Fun¸c˜oes Trigonom´etricas • cos x dx = sen x + c • sen x dx = − cos x + c • tg x dx = ln |sec x| + c • csc x dx = ln |csc x − cot x| + c • sec x dx = ln |sec x + tg x| + c • cot x dx = ln |sen x| + c • sec x tg x dx = sec x + c • csc x cot x dx = − csc x + c • sec2 x dx = tg x + c • csc2 x dx = − cot x + c • sen2 x dx = 1 2 (x − sen x cos x) + c • cos2 x dx = 1 2 (x + sen x cos x) + c Fun¸c˜oes Hiperb´olicas • sinh x dx = cosh x + c • cosh x dx = sinh x + c • tgh x dx = ln(cosh x) + c • csch x dx = ln tgh x 2 + c • sech x dx = arctg(sinh x) + c • coth x dx = ln | sinh x| + c 2
  • 3. Identidades Trigonom´etricas 1. sen(90o − θ) = cos θ 2. cos(90o − θ) = sen θ 3. sen θ cos θ = tg θ 4. sen2 θ + cos2 θ = 1 5. sec2 θ − tg2 θ = 1 6. csc2 θ − cot2 θ = 1 7. sen 2θ = 2 sen θ cos θ 8. cos 2θ = cos2 θ − sen2 θ = 2 cos2 θ − 1 9. sen 2θ = 2 sen θ cos θ 10. sen(α ± β) = sen α cos β ± cos α sen β 11. cos(α ± β) = cos α cos β sen α sen β 12. tg(α ± β) = tg α ± tg β 1 tg α tg β 13. sen α ± sen β = 2 sen 1 2 (α ± β) cos 1 2 (α ± β) 14. cos α + cos β = 2 cos 1 2 (α + β) cos 1 2 (α − β) 15. cos α − cos β = 2 sen 1 2 (α + β) sen 1 2 (α − β) 3