STEP II 2007 question 3 solution

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Part (a)

This gives:

Noting that , substitution gives:

So the integral is:

Q.E.D

Part (b)

Make the substitution which gives:

Part (c)

As indicated use the following substitution:

giving

The following identities are used:

and

The integral becomes:

Part (d)

Use the same method as in part (c) to show that this integral becomes: