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
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
Now lets deal with the limits:
Similarly for is
Now the integral becomes:
Now, we notice that this is a special case of the equation:
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 we should replace with
Now lets evaluate the limits for
Now the second integral becomes:
And we solve the problem!