I have solved a STEP problem from Stephen Siklos’ book *Advanced Problems in Core Mathematics*. The problem and the solution follows:

**Problem: **Use the substitution to evaluate

Hence or otherwise show that for

Where and

**Solution:**

At the first look it seems to me that solving the first part may give an idea of the second part. So lets begin with the first part of the question.

Lets substitute with

As

Now lets deal with the limits:

for

becomes

Similarly for is

Now the integral becomes:

Now, we notice that this is a *special case* of the equation:

Where and

Now we turn to the *second* part of the problem. I think if we could express the fraction as then we would find the solution.In order to do so we should find a number for which

So

And

So we should replace with

As

So

Now lets evaluate the limits for

For is

Similarly for

Now the second integral becomes:

And we solve the problem!

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## About f.nasim

A CS undergrad from Bangladesh.

Yes, your solution is correct. no problems i guess.