∫L
−L
cosmπxL sinnπxL dx= 0 for all integerm,nvalues
∫L
−L
cosmπxL cosnπxL dx=
0 , m 6 =n
L
2 , m=n^6 = 0
L, m=n= 0
∫∞
0
xmdx
1 + 2xcosβ+x^2 =
π
sinmπ
sinmβ
sinβ
∫∞
0
sinαx
x dx=
π/ 2 , α > 0
0 , α= 0
−π/ 2 , α < 0
∫∞
0
sinαxsinβx
x dx=
0 , α > β > 0
π/ 2 , 0 < α < β
π/ 4 , α=β > 0
∫∞
0
sinαxsinβx
x^2 dx=
{πα
2 ,^0 < α≤β
πβ
2 , α≥β >^0
∫∞
0
sin^2 αx
x^2 dx=
πα
2
∫∞
0
1 −cosαx
x^2 dx=
πα
2
∫∞
0
cosαx
x^2 +a^2 dx=
π
2 ae
−αa
∫∞
0
xsinαx
x(x^2 +a^2 )dx=
π
2 e
−αa
∫∞
0
sinx
xp dx=
π
2Γ(p) sin(pπ/2)
∫∞
0
cosx
xp dx=
π
2Γ(p) cos(pπ/2)
∫∞
0
tanx
x dx=
π
2
∫∞
0
sinαx
x(x^2 +a^2 )dx=
π
2 a^2 (1−e
−αa)
∫∞
0
sin^2 x
x^2 dx=
π
2
∫∞
0
sin^3 x
x^3 dx=
3 π
8
∫∞
0
sin^4 x
x^4 dx=
π
3
Appendix C
