
1450 INDEFINITE AND DEFINITE INTEGRALS
27.
Z
e
ax
sin bx dx = e
ax
a
a
2
+ b
2
sin bx −
b
a
2
+ b
2
cos bx
.
28.
Z
e
ax
sin
2
x dx =
e
ax
a
2
+ 4
a sin
2
x − 2 s in x cos x +
2
a
.
29.
Z
e
ax
sin
n
x dx =
e
ax
sin
n−1
x
a
2
+ n
2
(a sin x − n cos x) +
n(n − 1)
a
2
+ n
2
Z
e
ax
sin
n−2
x dx.
◮ Integrals involving sin x and cos x.
1.
Z
sin ax cos bx dx = −
cos[(a + b)x]
2(a + b)
−
cos
(a −b)x
2(a − b)
, a 6= ±b.
2.
Z
dx
b
2
cos
2
ax + c
2
sin
2
ax
=
1
abc
arctan
c
b
tan ax
.
3.
Z
dx
b
2
cos
2
ax − c
2
sin
2
ax
=
1
2abc
ln
c tan ax + b
c tan ax − b
.
4.
Z
dx
cos
2n
x sin
2m
x
=
n+m−1
X
k=0
C
k
n+m−1
tan
2k− 2m+1
x
2k − 2m + 1
, n, m = 1, 2, . . .
5.
Z
dx
cos
2n+1
x sin
2m+1
x
=C
m
n+m
ln |tan x|+
n+m
X
k=0
C
k
n+m
tan
2k− 2m
x
2k−2m
, n, m = 1, 2, . . .
◮ Reduction formulas.
⋆
The parameters ...