TABELA: Derivadas, Integrais
e Identidades Trigonom´etricas
•Derivadas
Sejam uevfun¸c˜oes deriv´aveis de xencon-
stante.
1. y=un⇒y0=n un−1u0.
2. y=uv ⇒y0=u0v+v0u.
3. y=u
v⇒y0=u0v−v0u
v2.
4. y=au⇒y0=au(ln a)u0,(a > 0, a 6= 1).
5. y=eu⇒y0=euu0.
6. y= logau⇒y0=u0
ulogae.
7. y= ln u⇒y0=1
uu0.
8. y=uv⇒y0=v uv−1u0+uv(ln u)v0.
9. y= sen u⇒y0=u0cos u.
10. y= cos u⇒y0=−u0sen u.
11. y= tg u⇒y0=u0sec2u.
12. y= cotg u⇒y0=−u0cosec2u.
13. y= sec u⇒y0=u0sec utg u.
14. y= cosec u⇒y0=−u0cosec ucotg u.
15. y=arc sen u⇒y0=u0
√1−u2.
16. y=arc cos u⇒y0=−u0
√1−u2.
17. y=arc tg u⇒y0=u0
1+u2.
18. y=arc cot g u ⇒−u0
1+u2.
19. y=arc sec u, |u|>1
⇒y0=u0
|u|√u2−1,|u|>1.
20. y=arc cosec u, |u|>1
⇒y0=−u0
|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 2x
2.
6. sen 2x= 2 sen xcos x.
7. 2 sen xcos y= sen (x−y) + sen (x+y).
8. 2 sen xsen y= cos (x−y)−cos (x+y).
9. 2 cos xcos y= cos (x−y) + cos (x+y).
10. 1 ±sen x= 1 ±cos ¡π
2−x¢.
•Integrais
1. Rdu =u+c.
2. Rundu =un+1
n+1 +c, n 6=−1.
3. Rdu
u= ln |u|+c.
4. Raudu =au
ln a+c, a > 0, a 6= 1.
5. Reudu =eu+c.
6. Rsen u du =−cos u+c.
7. Rcos u du = sen u+c.
8. Rtg u du = ln |sec u|+c.
9. Rcotg u du = ln |sen u|+c.
10. Rsec u du = ln |sec u+ tg u|+c.
11. Rcosec u du = ln |cosec u−cotg u|+c.
12. Rsec utg u du = sec u+c.
13. Rcosec ucotg u du =−cosec u+c.
14. Rsec2u du = tg u+c.
15. Rcosec2u du =−cotg u+c.
16. Rdu
u2+a2=1
aarc tg u
a+c.
17. Rdu
u2−a2=1
2aln ¯
¯
¯
u−a
u+a¯
¯
¯+c, u2> a2.
18. Rdu
√u2+a2= ln ¯
¯
¯u+√u2+a2¯
¯
¯+c.
19. Rdu
√u2−a2= ln ¯
¯
¯u+√u2−a2¯
¯
¯+c.
20. Rdu
√a2−u2=arc sen u
a+c, u2< a2.
21. Rdu
u√u2−a2=1
aarc sec ¯
¯u
a¯
¯+c.
•F´ormulas de Recorrˆencia
1.Rsennau du =−senn−1au cos au
an
+¡n−1
n¢Rsenn−2au du.
2. Rcosnau du =sen au cosn−1au
an
+¡n−1
n¢Rcosn−2au du.
3. Rtgnau du =tgn−1au
a(n−1) −Rtgn−2au du.
4. Rcotgnau du =−cotgn−1au
a(n−1) −Rcotgn−2au du.
5. Rsecnau du =secn−2au tg au
a(n−1)
+³n−2
n−1´Rsecn−2au du.
6. Rcosecnau du =−cosecn−2au cotg au
a(n−1)
+³n−2
n−1´Rcosecn−2au du.
