# Integration by substitution

Integration by substitution

- Created by: Nicola
- Created on: 24-06-10 01:31

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Week 8

W8.1 Integration by substitution

This is effectively the reverse of the Chain Rule for differentiation.

e.g. try integrating .

The problem is that you can't divide the bottom line into the top line.

But, if we could replace the bottom line by a single character,

it would be easy.

So put z =

Then = 2x.

If you now pretend that is a fraction, you can multiply both sides by

dx: dz = 2x dx

and so get a substitution for dx: dx =

Making substitutions for and for dx, make the integral into

.

Clearly, the x's cancel, giving you a straightforward z integral: .

Once you've performed that integration,

you reverse the substitution so as to finish with a function of x, not of z.

Work through what the book has to say before Exercise 8e

and then do number 1 from the exercise.

W8.2 Integration by substitution

The routine is slightly different for a definite integral.

Suppose you have

You do the same substitution as before,

but now you need to substitute for the limits of integration:

z(1) = 1 + 4 = 5

z(2) = 4 + 4 = 8

So the integral becomes

As this results in a number [with no z's in it], there's no need to undo the

substitution at the end.

Now do question 2 from Exercise 8e

and the Mixed Exercise on page 150.

CT NHC 06/11/2009

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