Understanding Engineering Mathematics

9.2.5 Linear substitution in integration

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It may be thatf(x)in

∫

f(x)dxis inconvenient for integration because of the variable

x, and it may be useful to change to a new variable. This entails changingdxas well as
f(x)of course. For example, consider the integral
∫
( 2 x− 4 )^3 dx

You may be tempted to expand the bracket out by the binomial theorem, and obtain a
polynomial which is easily integrated. This is correct, but is an unnecessary complication.
The way to go is to notice the similarity tox^3 and substitute

u= 2 x− 4

We must of course also replacedx. To this end we note that

du

dx

= 2

and sodu= 2 dxanddx=

du

2

. This somewhat cavalier way of dealing with thedxand

duis permissible provided we are careful. The upshot is that
∫
( 2 x− 4 )^3 dx=

∫

u^3

du

2

=

1

2

∫

u^3 du=

u^4

8

+C

=

( 2 x− 4 )^4

8

+C

on returning to the original variable.
In general, whenever we make a substitution of the form

u=ax+b

wherea,bare constants, we call this alinear substitution. It is particularly simple because

du

dx

=a

and thereforeduanddxare simply proportional

du=

du

dx

dx=adx

so

dx=

du

a

With practice, you may be able to dispense with the formal ‘u=x+1’ substitution and
do the above example as follows: