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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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W8.3 Integration by parts
This is a re-arrangement of the Product Rule for differentiation.
Work through the textbook's introduction of Exercise 9a .
Remember: to make the integration easier,
you should put u = something which gets easier when you differentiate
[often a power of x]
and = something which doesn't get worse when you integrate
[most often an exponential or trig function].…read more