Tabela de Derivadas, Integrais e Identidades Trigonométricas PDF

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Tabela com as principais derivadas, integrais e identidades trigonométricas....

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UNIVERSIDADE FEDERAL DO ABC Tabela de Derivadas, Integrais e Identidades Trigonom´etricas

Derivadas Regras de Deriva¸ca˜ o

Fun¸coes ˜ Trigonom´etricas Inversas

• (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¸coes ˜ Simples •

d c dx

=0

•

d x dx

=1

•

d cx dx

=c

d xc dx

Fun¸coes ˜ Exponenciais e Logar´ıtmicas •

d ex dx

•

d dx

•

d ax dx

=

d dx

arcsen x =

•

d dx

arccos x =

•

d dx

arctg x =

•

d dx

arcsec x =

•

d dx

arccotg x =

•

d dx

arccossec x =

√ −1 1−x2 1 1+x2 √1 |x| x2 −1 −1 1+x2 −1 √ | x| x2 −1

•

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¸coes ˜ Hiperbolicas ´ Inversas

ex

ln(x) =

√ 1 1− x 2

Fun¸coes ˜ Hiperbolicas ´

= cxc−1    −1  d 1 d = −x−2 = −x12 • dx x = dx x 1 c d d (x−c ) = − c+1 • dx xc = dx x d √ 1 d x 12 = 1 x− 21 = √ • dx , x = dx 2 2 x •

•

1 x

= ax ln(a)

Fun¸coes ˜ Trigonom´etricas

•

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 =

•

d dx

arcsech x =

√−1 x 1− x 2 1 1−x2

•

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 2 x

•

d dx

arccoth x =

•

d dx

cossec x = −cossec x cotg x

•

d dx

arccossech x =

1

1 1−x2

−1 √ |x| 1+x2

Integrais

Regras de Integra¸ca˜ o R cf(x) dx = c f(x) dx R R R • [f(x) + g(x)] dx = f(x) dx + g(x) dx R R • f ′ (x)g(x) dx = f(x)g(x) − f(x)g ′ (x) dx

•

R

Fun¸coes ˜ Racionais •

R

xn dx =

xn+1 n+1

+c

•

Z

1 dx = ln |x| + c x

•

Z

du = arctg u + c 1 + u2

Z

• •

R

•

Z

x −1 √ dx = arccos + c 2 2 a a −x

• •

=

• • •

ln x dx = x ln x − x + c x ln a

•

Z

du √ = arcsen u + c 1 − u2

•

Z

du √ = arcsec u + c u u2 − 1

•

+c

• • •

R

cos x dx = sen x + c

R

tg x dx = ln |sec x| + c

R

sec x dx = ln |sec x + tg x| + c

R

sen x dx = − cos x + c

R

csc x dx = ln |csc x − cot x| + c

R

cot x dx = ln |sen x| + c

R

csc x cot x dx = − csc x + c

R

sec x tg x dx = sec x + c

R

sec2 x dx = tg x + c

R

sen2 x dx = 12 (x − sen x cos x) + c

R

csc2 x dx = − cot x + c

R

cos2 x dx =

1 (x + sen x cos x) + c 2

Fun¸coes ˜ Hiperbolicas ´ •

du

√ = arcsenh u + c 1 + u2√ = ln |u + u2 + 1| + c Z du • √ = arccosh u + c 1 − u2√ = ln |u + u2 − 1| + c Z du • √ = −arcsech |u| + c u 1 − u2

•

1 x √ dx = arcsen + c 2 2 a a −x

•

Fun¸coes ˜ Irracionais

Z

Z

•

a2

loga x dx = x loga x −

•

•

Fun¸coes ˜ Logar´ıtmicas R

du √ = −arccosech |u| + c u 1 + u2

Fun¸coes ˜ Trigonom´etricas

para n 6= −1

1 1 dx = arctg(x/a) + c 2 a +x  Z du arctgh u + c, se |u| < 1 = • arccotgh u + c, se |u| > 1 1 − u2  1 ln  1+u  + c 1−u 2 •

•

Z

•

R

R

R

sinh x dx = cosh x + c cosh x dx = sinh x + c

tgh x dx = ln(cosh x) + c   R • csch x dx = ln tgh x2  + c R • sech x dx = arctg(sinh x) + c R • coth x dx = ln | sinh x| + c •

2

Identidades Trigonom´etricas 1. sen(90o − θ) = cos θ

9. sen 2θ = 2 sen θ cos θ

2. cos(90o − θ) = sen θ 3.

10. sen(α ± β) = sen α cos β ± cos α sen β

sen θ = tg θ cos θ

11. cos(α ± β) = cos α cos β ∓ sen α sen β 12. tg(α ± β) =

4. sen2 θ + cos2 θ = 1 5. sec2 θ − tg2 θ = 1

tg α ± tg β 1 ∓ tg α tg β

13. sen α ± sen β = 2 sen

6. csc2 θ − cot2 θ = 1

1 1 (α ± β) cos (α ± β) 2 2

1 1 14. cos α + cos β = 2 cos (α + β) cos (α − β) 2 2

7. sen 2θ = 2 sen θ cos θ

1 1 15. cos α − cos β = 2 sen (α + β) sen (α − β) 2 2

8. cos 2θ = cos2 θ − sen2 θ = 2 cos2 θ − 1

3...