In my calculus class in university, we learned how to use substitution to solve integrals. But then on the quiz we got this question:
Find the numerical value of the following integral:
$$\int_0^\frac{\pi}{2} \frac{{\sin(x)}^{22222222}}{{\sin(x)}^{22222222} + {\cos(x)}^{22222222}} dx$$
Hint: Use the substitution $u = \frac{\pi}{2} - x$.
Second hint: You cannot find an elementary anti-derivative of this.
Give your answer with 3 decimal digits.
Recommended time: 10 minutes
I could not understand it, even when the teacher explained it after the quiz. He was saying that it is the same for every power of the form $2n$ where $n$ is a natural number.
What should I do to solve this?