- IfSm=
∫
xmsinnx dxandCm=
∫
xmcosnx dx, then
Sm=−n^1 xmcosnx+mnCm− 1 and Cm=^1 nxmsinnx−mnSm− 1
- IfI 1 =
∫
tanx dx, andIn=
∫
tannx dx, then In=n−^11 tann−^1 x−In− 2 , n= 2, 3 , 4 ,...
- IfIn=
∫ sinnax
cosaxdx, thenIn=−
sinn−^1 ax
(n−1)a +In−^2
- IfIn=
∫ cosnax
sinax dx, thenIn=
cosn−^1 ax
(n−1)a +In−^2
- IfIn,m=
∫
sinnxcosmx dx, then
In,m=n−+^1 msinn−^1 xcosm+1x+nn+−m^1 In− 2 ,m
In,m=n+ 1^1 sinn+1xcosm+1x+n+nm+ 1+ 2In+2,m
In,m=n+^1 msinn+1xcosm−^1 x+mn+−m^1 In,m+2
In,m=m−+ 1^1 sinn+1xcosm+1x+n+mm+ 1+ 2In,m+2
In,m=m−+ 1^1 sinn−^1 xcosm+1x+mn−+ 1^1 In− 2 ,m+2
In,m=n+ 1^1 sinn+1xcosm−^1 x+mn+ 1−^1 In+2,m− 2
- IfSn=
∫
eaxsinnbx dxandCn=
∫
eaxcosnbx dx, then
Cn=eaxcosn−^1 bx
[
acosbx+nbsinbx
a^2 +n^2 b^2
]
+n(n−1)b
2
a^2 +n^2 b^2 Cn−^2
Sn=eaxsinn−^1 ax
[
asinbx−nbcosnx
a^2 +n^2 b^2
]
+n(n−1)b
2
a^2 +n^2 b^2 Sn−^2
- IfIn=
∫
xm(lnx)ndx, thenIn=m^1 + 1xm+1(lnx)n−mn+ 1In− 1
Integrals involving Bessel functions
∫
J 1 (x)dx=−J 0 (x) +C
∫
xJ 1 (x)dx=−xJ 0 (x) +
∫
J 0 (x)dx
∫
xnJ 1 (x)dx=−xnJ 0 (x) +n
∫
xn−^1 J 0 (x)dx
Appendix C
