problem 2 solution: The weird integral solution

So this was the one I made up. I'm kind of proud of it because so far, no one in my family has been able to figure it out :). The solution is actually not a hard one.

In fact, it's really easy to do once you recall the derivative of  inverse tangent.

looks familiar doesn't it? All I really did was multiply the derivative of arctan(e^x) with the function itself. And I love how no one was able to figure it out.

Solution

We just have to follow the standard procedure for u-substitution and we're good.

When we differentiate both sides it looks as follows. Notice how it looks almost exactly like the function we started out with.

We now have all the ingredients for u-substitution.