Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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.