Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

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.

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