A Very Challenging Integral

Calculus Challenge

Today’s Challenge

This is the third integral I have looked at from my old calculus textbook.

Give the problem a try before jumping in for the solution!

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Solution

I will first establish useful results that allow us to break down the integral.

We are given that

We apply the substitution

to the following integral

We can transform the above integral as follows

We can rewrite u as x, f(x) as f(u) as u is just an arbitrary variable.

The result is therefore

To help solve our integral, we will look at a slightly simpler version of it

Notice the lower bound has changed from 0 to pi/2 and the x in the numerator is eliminated.

We will apply the following substitution

And show the following

To write out explicitly, we have

I have replaced us and xs as u is just an arbitrary variable

Now, we know that

This means the integral

Merging the two integrals on the right, we have

Using the results established above, we have

Well done!

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Happy reading,

Barry 🍩