Integrales

Apéndice B
Integrales
B.1
Fórmulas más usadas
ˆ
ˆ
un du =
un+1
+ c; n �= −1
n+1
n+1
n
(au + b) du =
(B.3)
1
1
du = ln (au + b) + c
au + b
a
(B.4)
ˆ
ˆ
ˆ
ueau du =
ˆ
1 au
e +c
a
(B.5)
1 au
e (au − 1) + c
a2
(B.6)
eau du =
un eau du =
ˆ
(B.2)
1
du = lnu + c
u
ˆ
ˆ
(au + b)
+ c; n �= −1
a (n + 1)
(B.1)
n
un eau
−
a
a
ˆ
un−1 eau du
(B.7)
ln udu = u ln u − u
(B.8)
1
sen (au) du = − cos (au) + c
a
(B.9)
391
392
APÉNDICE B. INTEGRALES
ˆ
B.2
cos (au) du =
1
sen (au) + c
a
(B.10)
ˆ
1
tg (au) du = − ln cos (au) + c
a
(B.11)
ˆ
ctg (au) du =
1
ln sen (au) + c
a
(B.12)
1
eau [b sen (bu) + a cos (bu)] + c
+ b2
(B.13)
1
eau [a sen (bu) − b cos (bu)] + c
a 2 + b2
(B.14)
ˆ
eau cos (bu) du =
ˆ
eau sen (bu) du =
a2
Ortogonalidad y paralelismo
ˆ2π
ˆ2π
cos (mx) cos (nx) dx =
sen (mx) sen (nx) dx =
ˆ2π
ˆ2π


π;


0;


π;


0;
m=n
(B.15)
m �= n
m=n
(B.16)
m �= n
sen (mx) cos (nx) dx = 0
(B.17)


2π ;
(B.18)
cos (nx) dx =
ˆ2π


0;
n=0
n = 1, 2, . . .
sen (nx) dx = 0
(B.19)
B.2. ORTOGONALIDAD Y PARALELISMO
ˆ2π
ˆ2π
ˆ2π
ˆ2π
393
cos2 (nx) dx = π
(B.20)
sen2 (nx) dx = π
(B.21)
ejkx dx =


2π;


0;
ejmx e−jnx dx =
k=0
(B.22)
k = 1, 2, · · ·


2π;


0;
m=n
(B.23)
m �= n
394
APÉNDICE B. INTEGRALES