Integral Calculus Really need help

Question

Let n ∈ N. prove that integral (from 0 to pi/2) [sinⁿ(x)/sinⁿ(x)+cosⁿ(x)]dx = pi/4

please i really need help this is the only question i can't answer..

Answer 1

If you use a tool like Wolfram Alpha to solve this when n = 2 or 3, the solution proposed is very complicated. Here are some hints. I think if you multiply the original integrand (numerator and denominator) by csc⁽ⁿ⁺²⁾x you can transform the integrand into something like:

∫ csc²x/(csc²x + csc²xcot²x) dx.

That can be transformed into ∫ 1/(1+cot²x) dx (Is that step legitimate, since csc²x at 0 is not defined?). Something to think about.

Finally, with a bit of manipulation: ∫ sin²x dx. From 0 to π/2 that evaluates to π/4. Try the following graph and apply the slider to set n to various values.

https://www.desmos.com/calculator/rxrosty9bv

You still need to determine the antiderivative for sin²x and likely make some observations about what is happening at 0, but this should give you some ideas.