Integral bounds Given an integral like the following:
$\int_0^1 x^n (1-x)^n dx$ and the substitution like $x=\sin^2 \theta$
What would the limits be?
I went to proceed like so:
$1=\sin^2 \theta$ or $\cos^2 \theta = 0$ so $\theta = \pm \frac{\pi}{2}+2n\pi$
Also, for the lower bound: $0=\sin^2 \theta$ or $\theta = \frac{n\pi}{2}$.
How do I know choose the bounds?

By Euler's beta function,
$$ \int_{0}^{1}x^n(1-x)^n\,dx = \frac{\Gamma(n+1)^2}{\Gamma(2n+2)} = \frac{1}{(2n+1)\binom{2n}{n}}.\tag{1}$$
$(1)$ may be checked by integration by parts, no need of trigonometric substitutions.
