Understanding Engineering Mathematics

We consider changes of variable,x, later under various substitution methods, but here we
will have a first look at what we might be able to do with the integrand in some simple
cases. In generalf(x)may be a polynomial or rational function, trig or hyperbolic, log
or exponential, or any combination of these. We would first look atf(x)to see if it can
be simplified or rearranged to our advantage. This involves all that we have learnt so far,
including such things as

• algebraic simplification


• partial fractions


• trig identities


• exponential and log properties

In this section we concentrate on the simplest kinds of rearrangements. For example,∫

elnxdxlooks awful – but it is simply

∫

xdx, a much easier proposition. The review

question gives further examples.

Solution to review question 9.1.4

All the integrals look formidable (deliberately!) – but in fact, using the

properties of exponentials and logs and trig identities the integrands can

all be simplified to easily integrated functions.

(i)

∫

dx

cos 2x+2sin^2 x

=

∫

dx

1 −2sin^2 x+2sin^2 x

using cos 2x= 1 −2sin^2 x

=

∫

dx

1

=x+C

(ii)

∫

xe2lnxdx=

∫

xelnx

2

(αlnx=lnxα)

=

∫

xx^2 dx(elnx=x)=

∫

x^3 dx

=

x^4

4

+C

Note that here, as elsewhere, we have taken the modulus under the

log for granted, as to include it would clutter up the expressions.

(iii) This is easy if you know your compound angle formulae backwards!

∫

(cos 2xsin 3x−cos 3xsin 2x)dx=

∫

sin( 3 x− 2 x)dx

=

∫

sinxdx

=−cosx+C