Tabela completa de derivadas e integrais

Tabela para as matérias de Cálculo, derivadas, várias funções úteis e regras de integração e derivação.

1. UNIVERSIDADE FEDERAL DO ABC
Tabela de Derivadas, Integrais e Identidades Trigonom´etricas
Derivadas
Regras de Deriva¸c˜ao
(cf(x))
0
=cf
0
(x)
Derivada da Soma
(f(x) +g(x))
0
=f
0
(x) +g
0
(x)
Derivada do Produto
(f(x)g(x))
0
=f
0
(x)g(x) +f(x)g
0
(x)
Derivada do Quociente

f(x)
g(x)

0
=
f
0
(x)g(x) -f(x)g
0
(x)
g(x)
2
Regra da Cadeia
(f(g(x))
0
= (f
0
(g(x))g
0
(x)
Fun¸c˜oes Simples

d
dx
c=0

d
dx
x=1

d
dx
cx=c

d
dx
x
c
=cx
c-1

d
dx

1
x

=
d
dx

x
-1

= -x
-2
= -
1
x
2

d
dx

1
x
c

=
d
dx
(x
-c
)= -
c
x
c+1

d
dx
p
x=
d
dx
x
1
2=
1
2
x
-
1
2=
1
2
p
x
,
Fun¸c˜oes Exponenciais e Logar´tmicas

d
dx
e
x
=e
x

d
dx
ln(x) =
1
x

d
dx
a
x
=a
x
ln(a)
Fun¸c˜oes Trigonom´etricas

d
dx
senx=cosx

d
dx
cosx= -senx,

d
dx
tgx=sec
2
x

d
dx
secx=tgxsecx

d
dx
cotgx= -cossec
2
x

d
dx
cossecx= -cossecxcotgx
Fun¸c˜oes Trigonom´etricas Inversas

d
dx
arcsenx=
1
p
1-x
2

d
dx
arccosx=
-1
p
1-x
2

d
dx
arctgx=
1
1+x
2

d
dx
arcsecx=
1
jxj
p
x
2
-1

d
dx
arccotgx=
-1
1+x
2

d
dx
arccossecx=
-1
jxj
p
x
2
-1
Fun¸c˜oes Hiperb´olicas

d
dx
senhx=coshx=
e
x
+e
-x
2

d
dx
coshx=senhx=
e
x
-e
-x
2

d
dx
tghx=sech
2
x

d
dx
sechx= -tghxsechx

d
dx
cotghx= -cossech
2
x
Fun¸c˜oes Hiperb´olicas Inversas

d
dx
cschx= -cothxcossechx

d
dx
arcsenhx=
1
p
x
2
+1

d
dx
arccoshx=
1
p
x
2
-1

d
dx
arctghx=
1
1-x
2

d
dx
arcsechx=
-1
x
p
1-x
2

d
dx
arccothx=
1
1-x
2

d
dx
arccossechx=
-1
jxj
p
1+x
2
1

2. Integrais
Regras de Integra¸c˜ao

R
cf(x)dx=c
R
f(x)dx

R
[f(x) +g(x)]dx=
R
f(x)dx+
R
g(x)dx

R
f
0
(x)g(x)dx=f(x)g(x) -
R
f(x)g
0
(x)dx
Fun¸c˜oes Racionais

R
x
n
dx=
x
n+1
n+1
+c paran6= -1

Z
1
x
dx=lnjxj+c

Z
du
1+u
2
=arctgu+c

Z
1
a
2
+x
2
dx=
1
a
arctg(x=a) +c

Z
du
1-u
2
=

arctghu+c, sejuj< 1
arccotghu+c, sejuj> 1
=
1
2
ln

1+u
1-u


+c
Fun¸c˜oes Logar´tmicas

R
lnx dx=xlnx-x+c

R
log
a
x dx=xlog
a
x-
x
lna
+c
Fun¸c˜oes Irracionais

Z
du
p
1-u
2
=arcsenu+c

Z
du
u
p
u
2
-1
=arcsecu+c

Z
du
p
1+u
2
=arcsenhu+c
=lnju+
p
u
2
+1j+c

Z
du
p
1-u
2
=arccoshu+c
=lnju+
p
u
2
-1j+c

Z
du
u
p
1-u
2
= -arcsechjuj+c

Z
du
u
p
1+u
2
= -arccosechjuj+c

Z
1
p
a
2
-x
2
dx=arcsen
x
a
+c

Z
-1
p
a
2
-x
2
dx=arccos
x
a
+c
Fun¸c˜oes Trigonom´etricas

R
cosx dx=senx+c

R
senx dx= -cosx+c

R
tgx dx=lnjsecxj+c

R
cscx dx=lnjcscx-cotxj+c

R
secx dx=lnjsecx+tgxj+c

R
cotx dx=lnjsenxj+c

R
secxtgx dx=secx+c

R
cscxcotx dx= -cscx+c

R
sec
2
x dx=tgx+c

R
csc
2
x dx= -cotx+c

R
sen
2
x dx=
1
2
(x-senxcosx) +c

R
cos
2
x dx=
1
2
(x+senxcosx) +c
Fun¸c˜oes Hiperb´olicas

R
sinhx dx=coshx+c

R
coshx dx=sinhx+c

R
tghx dx=ln(coshx) +c

R
cschx dx=ln


tgh
x
2


+c

R
sechx dx=arctg(sinhx) +c

R
cothx dx=lnjsinhxj+c
2

3. Identidades Trigonom´etricas
1. (90
o
-) =cos
2. (90
o
-) =sen
3.
sen
cos
=tg
4.
2
+cos
2
=1
5.
2
-tg
2
=1
6.
2
-cot
2
=1
7. 2=2sencos
8. 2=cos
2
-sen
2
=2cos
2
-1
9. 2=2sencos
10. () =sencoscossen
11. () =coscossensen
12. () =
tgtg
1tgtg
13. sen=2sen
1
2
()cos
1
2
()
14. +cos=2cos
1
2
(+)cos
1
2
(-)
15. -cos=2sen
1
2
(+)sen
1
2
(-)
3