TABELA: Derivadas, Integrais e Identidades Trigonom ́etricas

TABELA: Derivadas, Integrais e Identidades Trigonom ́etricas

• Derivadas

Sejam u e v fun ̧co ̃es deriv ́aveis de x e n con- stante.

1. y=un ⇒y′ = nun−1u′. 2. y=uv ⇒y′ =u′v+v′u.

u ′ u′v−v′u 3.y=v⇒y=v2 .

4. y=au ⇒y′ =au(lna)u′, (a>0, a̸=1).

5. y=eu ⇒y′ =euu′.

• Integrais 1. 􏰁 du=u+c.

2.􏰁undu=un+1 +c,n̸=−1. 􏰁 du n+1

3.􏰁 u =ln|u|+c.

4. audu= au +c, a>0, a̸=1.

􏰁lna

5. 􏰁eudu=eu+c.

6.y=log u ⇒y′=u′ log e. aua

􏰁

6. 􏰁 senudu=−cos u+c. 7. cos udu=senu+c.

7.y=lnu ⇒y′=1u′. u

8. tgudu=ln|sec u|+c. 􏰁

9. cotg u du = ln |sen u| + c.

8.y=uv ⇒y′=vuv−1u′+uv(lnu)v′. 9. y=senu ⇒y′ =u′cos u.

10. y=cos u ⇒y′ =−u′senu.

11. y=tgu ⇒y′ =u′sec2u.

12. y = cotg u ⇒ y′ = −u′cosec2u.

13. y=sec u ⇒y′ =u′sec utgu.

14. y=cosecu ⇒y′ =−u′cosecucotgu.

15.y=arcsenu ⇒y′=√u′ . 1−u2

′ −u′ 16.y=arccosu ⇒y =√1−u2.

′

−u′ 18.y=arccotgu ⇒1+u2.

10. 􏰁 sec udu=ln|sec u+tgu|+c.

11. 􏰁 cosecudu=ln|cosecu−cotgu|+c.

12. 􏰁 sec utgudu=sec u+c.

13. 􏰁 cosecucotgudu=−cosecu+c.

14. 􏰁 sec2udu=tgu+c.

15. 􏰁cosec2udu=−cotgu+c.

16. 􏰁 du =1arctgu+c. 􏰁u2+a2a􏰄a􏰄

17. du =1 ln􏰄􏰄u−a􏰄􏰄+c,u2>a2.

u2 −a2 2a 􏰁􏰄􏰄√􏰄􏰄

u+a

17. y=arctgu ⇒y′ = u 2. 1+u

18. √ du =ln􏰄u+ u2+a2􏰄+c.

u2+a2 􏰄 􏰁√

􏰄

19. √ du =ln􏰄􏰄u+ u2−a2􏰄􏰄+c.

19. y=arc sec u, |u|􏰀1

⇒y′ = u′ ,|u|>1.

􏰁 u2−a2

20. √ du =arcsenu +c, u2 <a2.

|u|√u2 −1 20. y=arccosecu,|u|􏰀1

􏰁a2−u2 a􏰄􏰄

21.