TABELA: Derivadas, Integrais
e Identidades Trigonom´
etricas
•
Derivadas
Sejam u e v fun¸c˜oes deriv´aveis de x e n con-stante.
1. y =un ⇒y′ = n un−1u′. 2. y =uv ⇒y′ =u′v+v′u. 3. y = u
v ⇒y′ =
u′v−v′u
v2 .
4. y =au ⇒y′ =au(lna)u′, (a >0, a6= 1). 5. y =eu ⇒y′ =euu′.
6. y = logau ⇒y′ = u′
u logae.
7. y = lnu ⇒y′ = 1uu′.
8. y =uv ⇒y′ =v uv−1 u′+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= cotgu ⇒y′ =−u′cosec2u. 13. y= sec u ⇒y′ =u′sec utgu.
14. y= cosecu ⇒y′ =−u′cosecucotgu. 15. y=arcsenu ⇒y′ = √u′
1−u2.
16. y=arc cos u ⇒y′ = −u′
√ 1−u2.
17. y=arctgu ⇒y′ = u′
1+u2.
18. y=arc cotg u ⇒ 1+−uu′2.
19. y=arc sec u, |u|>1
⇒y′ = u′
|u|√u2−1,|u|>1.
20. y=arccosecu,|u|>1
⇒y′ = −u′
|u|√u2−1,|u|>1.
•
Identidades Trigonom´
etricas
1. sen2x+ cos2x= 1.
2. 1 + tg2x= sec2x. 3. 1 + cotg2x= cosec2x. 4. sen2x= 1−cos 2x
2 .
5. cos2x= 1+cos 22 x. 6. sen 2x= 2 senx cos x.
7. 2 senx cos y= sen (x−y) +sen(x+y). 8. 2 senxseny= cos (x−y)−cos (x+y). 9. 2 cos x cos y= cos (x−y) + cos (x+y). 10. 1±senx= 1±cos¡π
2 −x ¢
.
•
Integrais
1. R
du=u+c. 2. R
undu= unn+1+1 +c, n6=−1. 3. R du
u = ln|u|+c.
4. R
audu= au
lna +c, a >0, a6= 1.
5. R
eudu=eu+c. 6. R
senu du=−cos u+c. 7. R
cos u du= senu+c. 8. R
tgu du= ln|sec u|+c. 9. R
cotgu du= ln|senu|+c. 10. R
sec u du= ln|sec u+ tgu|+c. 11. R
cosecu du= ln|cosecu−cotgu|+c. 12. R
sec utgu du= sec u+c. 13. R
cosecucotgu du=−cosecu+c. 14. R
sec2u du= tgu+c.
15. R
cosec2u du=−cotgu+c. 16. R du
u2+a2 = 1aarctgua+c.
17. R du
u2−a2 =21aln
¯ ¯ ¯
u−a u+a
¯ ¯
¯+c, u
2> a2.
18. R du √
u2+a2 = ln
¯ ¯
¯u+
√
u2+a2
¯ ¯
¯+c.
19. R du √
u2
−a2 = ln
¯ ¯
¯u+
√
u2−a2¯¯
¯+c.
20. R du √
a2
−u2 =arcsen
u
a+c, u2< a2.
21. R du
u√u2−a2 =
1
aarc sec
¯ ¯ua
¯
¯+c.
•
F´
ormulas de Recorrˆ
encia
1.R
sennau du=−senn−1au cosau
an
+¡n−1
n
¢ R
senn−2au du.
2. R
cosnau du= sen au cosn−1au
an
+¡n−1
n
¢ R
cosn−2au du.
3. R
tgnau du= tga(nn−−11)au−R
tgn−2au du.
4. R
cotgnau du=−cotga(nn−−1)1au−R
cotgn−2au du.
5. R
secnau du= secn−2au tg au
a(n−1)
+³nn−−21´R
secn−2au du.
6. R
cosecnau du=−cosecna−(2nau cotg au−1) +³n−2
n−1 ´
R
