Understanding Engineering Mathematics

which enable us to integrate functions of the type

sinmxcosnx

sinmxsinmx (m   =n)

cosmxcosnx

Such integrals are fundamental to calculations in the theory of Fourier series (Chapter 17).
The double angle formulae can help us deal with integrals containing powers of sinxor
cosx. Odd powers of sinxand cosxcan sometimes be dealt with using the Pythagorean
identity cos^2 x+sin^2 x=1 and substitution. In general, for the integral:

∫

sinmxcosnxdx

if one ofmornis odd, try using the Pythagorean identity whilst if both are even, use
double angle formula and other trig identities. The review questions illustrate this.
For integrals such as
∫
cosmxcosnxdx,

∫

sinmxsinnxdx,

∫

sinmxcosnxdx

wherem =n, use the compound angle formulae. Again, see the review questions. Hyper-
bolic functions can be dealt with similarly.

Solution to review question 9.1.8

The use of trig identities for the examples given relies on the fact

that we can always integrate cos or sine of multiple angles:

∫

coskxdx=

1

k

sinkx+C

∫

sinkxdx=−

1

k

coskx+C

When faced with integrals such as those in this question, make a list

of the identities you know involving products of sines and cosines:

cos 2x=2cos^2 x− 1 = 1 −2sin^2 x

=cos^2 x−sin^2 x

sin 2x=2sinxcosx

sin(A+B)=sinAcosB+sinBcosA,etc.

using these, the integrals are not so bad:

(i)

∫

sin^2 xdx=

∫

1

2

( 1 −cos 2x)dx