Use $\int_0^{\pi/2}\frac{1}{1 + \cos x + \sin x}dx = \ln 2$ to find $\int_0^{\pi/2}\frac{x}{1 + \cos x + \sin x}dx$

Question

I am a PhD student studying analytic number theory, but a high school student showed me this integration problem and I couldn't quite crack it. I am wondering if anyone on here has some thoughts. The question is:

By using the fact that $$\int_0^{\pi/2}\frac{1}{1 + \cos x + \sin x}dx = \ln 2 $$

compute
$$\int_0^{\pi/2}\frac{x}{1 + \cos x + \sin x}dx$$

Of course, stick to tools that would be available to a first-year calculus student.

Answer 1

Hint: Deal with the second integral using the substitution $x=\frac\pi2-y$ and $\mathrm dx=-\mathrm dy$.