Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(lu) #1

2.2 INTEGRATION


Evaluate the integral

I=


1


x^2 +x

dx.

We note that the denominator factorises to givex(x+ 1). Hence


I=



1


x(x+1)

dx.

We now separate the fraction into two partial fractions and integrate directly:


I=


∫(


1


x


1


x+1

)


dx=lnx−ln(x+1)+c=ln

(


x
x+1

)


+c.

2.2.7 Integration by substitution

Sometimes it is possible to make a substitution of variables that turns a com-


plicated integral into a simpler one, which can then be integrated by a standard


method. There are many useful substitutions and knowing which to use is a matter


of experience. We now present a few examples of particularly useful substitutions.


Evaluate the integral

I=


1



1 −x^2

dx.

Making the substitutionx=sinu, we note thatdx=cosudu, and hence


I=



1



1 −sin^2 u

cosudu=


1



cos^2 u

cosudu=


du=u+c.

Now substituting back foru,


I=sin−^1 x+c.

This corresponds to one of the results given in subsection 2.2.3.


Another particular example of integration by substitution is afforded by inte-

grals of the form


I=


1
a+bcosx

dx or I=


1
a+bsinx

dx. (2.33)

In these cases, making the substitutiont= tan(x/2) yields integrals that can


be solved more easily than the originals. Formulae expressing sinxand cosxin


terms oftwere derived in equations (1.32) and (1.33) (see p. 14), but before we


can use them we must relatedxtodtas follows.

Free download pdf