Mathematical Methods for Physics and Engineering : A Comprehensive Guide

2.2 INTEGRATION

found near the end of subsection 2.1.1. A few are presented below, using the form

given in (2.30):

∫

adx=ax+c,

∫

axndx=

axn+1

n+1

+c,

∫

eaxdx=

eax

a

+c,

∫

a

x

dx=alnx+c,

∫

acosbx dx=

asinbx

b

+c,

∫

asinbx dx=

−acosbx

b

+c,

∫

atanbx dx=

−aln(cosbx)

b

+c,

∫

acosbxsinnbx dx=

asinn+1bx

b(n+1)

+c,

∫

a

a^2 +x^2

dx=tan−^1

(x

a

)

+c,

∫

asinbxcosnbx dx=

−acosn+1bx

b(n+1)

+c,

∫

− 1

√

a^2 −x^2

dx=cos−^1

(x

a

)

+c,

∫

1

√

a^2 −x^2

dx=sin−^1

(x

a

)

+c,

where the integrals that depend onnare valid for alln=−1andwhereaandb

are constants. In the two final results|x|≤a.

2.2.4 Integration of sinusoidal functions

Integrals of the type

∫

sinnxdxand

∫

cosnxdxmay be found by using trigono-

metric expansions. Two methods are applicable, one for oddnand the other for

evenn. They are best illustrated by example.

Evaluate the integralI=

∫

sin^5 xdx.

Rewriting the integral as a product of sinxand an even power of sinx, and then using
the relation sin^2 x=1−cos^2 xyields

I=

∫

sin^4 xsinxdx

=

∫

(1−cos^2 x)^2 sinxdx

=

∫

(1−2cos^2 x+cos^4 x)sinxdx

=

∫

(sinx−2sinxcos^2 x+sinxcos^4 x)dx

=−cosx+^23 cos^3 x−^15 cos^5 x+c,

where the integration has been carried out using the results of subsection 2.2.3.