tab-integrais
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tab-integrais


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TABELA: Derivadas, Integrais
e Identidades Trigonom´etricas
Derivadas
Sejam uevfun¸oes deriv´aveis de xencon-
stante.
1. y=uny0=n un1u0.
2. y=uv y0=u0v+v0u.
3. y=u
vy0=u0vv0u
v2.
4. y=auy0=au(ln a)u0,(a > 0, a 6= 1).
5. y=euy0=euu0.
6. y= logauy0=u0
ulogae.
7. y= ln uy0=1
uu0.
8. y=uvy0=v uv1u0+uv(ln u)v0.
9. y= sen uy0=u0cos u.
10. y= cos uy0=u0sen u.
11. y= tg uy0=u0sec2u.
12. y= cotg uy0=u0cosec2u.
13. y= sec uy0=u0sec utg u.
14. y= cosec uy0=u0cosec ucotg u.
15. y=ar c sen uy0=u0
1u2.
16. y=ar c cos uy0=u0
1u2.
17. y=ar c tg uy0=u0
1+u2.
18. y=ar c cot g u u0
1+u2.
19. y=ar c sec u, |u|>1
y0=u0
|u|u21,|u|>1.
20. y=ar c cosec u, |u|>1
y0=u0
|u|u21,|u|>1.
Identidades Trigonom´etricas
1. sen2x+ cos2x= 1.
2. 1 + tg2x= sec2x.
3. 1 + cotg2x= cosec2x.
4. sen2x=1cos 2x
2.
5. cos2x=1+cos 2x
2.
6. sen 2x= 2 sen xcos x.
7. 2 sen xcos y= sen (xy) + sen (x+y).
8. 2 sen xsen y= cos (xy)cos (x+y).
9. 2 cos xcos y= cos (xy) + cos (x+y).
10. 1 ±sen x= 1 ±cos ¡π
2x¢.
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 ucotg 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
aar c tg u
a+c.
17. Rdu
u2a2=1
2aln ¯
¯
¯
ua
u+a¯
¯
¯+c, u2> a2.
18. Rdu
u2+a2= ln ¯
¯
¯u+u2+a2¯
¯
¯+c.
19. Rdu
u2a2= ln ¯
¯
¯u+u2a2¯
¯
¯+c.
20. Rdu
a2u2=ar c sen u
a+c, u2< a2.
21. Rdu
uu2a2=1
aar c sec ¯
¯u
a¯
¯+c.
ormulas de Recorrˆencia
1.Rsennau du =senn1au cos au
an
+¡n1
n¢Rsenn2au du.
2. Rcosnau du =sen au cosn1au
an
+¡n1
n¢Rcosn2au du.
3. Rtg nau du =tg n1au
a(n1) Rtg n2au du.
4. Rcotg nau du =cotg n1au
a(n1) Rcotg n2au du.
5. Rsecnau du =secn2au tg au
a(n1)
+³n2
n1´Rsecn2au du.
6. Rcosecnau du =cosecn2au cotg au
a(n1)
+³n2
n1´Rcosecn2au du.