Definite Integrals

Question

The value of $\int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx$ is:

RankGuru · Accepted Answer

Let $I = \int_0^{\pi/2} \frac{\sin^{n} x}{\sin^{n} x + \cos^{n} x} dx$.
Using Property $\int_0^a f(x) dx = \int_0^a f(a-x) dx$:
Replace $x$ with $\pi/2 - x$. Note that $\sin(\pi/2 - x) = \cos x$ and $\cos(\pi/2 - x) = \sin x$.
$I = \int_0^{\pi/2} \frac{\cos^{n} x}{\cos^{n} x + \sin^{n} x} dx$.

Adding the two expressions for $I$:
$2I = \int_0^{\pi/2} \frac{\sin^n x + \cos^n x}{\sin^n x + \cos^n x} dx$
$2I = \int_0^{\pi/2} 1 dx = [x]_0^{\pi/2} = \pi/2$.
So, $I = \pi/4$.

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