∫∞
0
sinax^2 cos 2bx dx=^12
√
π
2 a
(
cosb
2
a −sin
b^2
a
)
∫∞
0
cosax^2 cos 2bx dx=^12
√
π
2 a
(
cosb
2
a + sin
b^2
a
)
∫∞
0
dx
x^4 + 2a^2 x^2 cos 2β+a^4 =
π
4 a^3 cosβ
∫∞
0
cos
(
x^2 +a
2
x^2
)
dx=
√π
2 cos(
π
4 + 2a)
∫∞
0
sin
(
x^2 +a
2
x^2
)
dx=
√π
2 sin(
π
4 + 2a)
∫∞
0
tanbx dx
x(p^2 +x^2 )=
π
2 p^2 tanhbp
∫∞
0
xtanbx dx
p^2 +x^2 =
π
2 −
π
2 tanhbp
∫∞
0
xcotbx dx
p^2 +x^2 =
π
2 cothbp
∫∞
0
sinax
sinbx
dx
(p^2 +x^2 )=
π
2 p
sinhap
sinhbp, a < b
∫∞
0
cosax
cosbx
dx
(p^2 +x^2 )=
π
2 p
coshap
coshbp, a < b
∫∞
0
sinax
cosbx
dx
(p^2 +x^2 )=
π
2 p^2
sinhap
coshbp, a < b
∫∞
0
sinax
cosbx
x dx
(x^2 +p^2 )=−
π
2
sinhap
coshbp, a < b
∫∞
0
cosax
sinbx
x dx
(p^2 +x^2 )=
π
2
coshap
sinhbp, a < b
Integrals containing exponential and logarithmic terms
∫ 1
0
ln^1 x
1 +xdx=
π^2
12
∫ 1
0
ln^1 x
(1−x)dx=
π^2
6
∫ 1
0
(
ln^1 x
) 3
1 −x dx=
π^4
15
∫ 1
0
ln(1 +x)
x dx=
π^2
12
Appendix C
