Integration by substitution of $\int 1/\sqrt{x(1-x)}\, dx$

Question

Find $$\int \frac {1}{\sqrt{x(1-x)}}\,dx$$ using the the substitution $x=\sin^2(u)$

Answer 1

Let $x = \sin^{2}(u)$ then $dx = 2\sin(u)\cos(u)\,du$, so the integral becomes: \begin{align*} \int \dfrac{2\sin(u)\cos(u)du}{\sqrt{\sin^2(u)(1-\sin^2(u))}} &= 2 \int \dfrac{\sin(u)\cos(u)\,du}{\sqrt{\sin^{2}(u)\cos^{2}(u)}} \\ &= 2 \int \dfrac{\sin(u)\cos(u)du}{\sin(u)\cos(u)} \\ &= 2 \int \,du \\ &= 2u +C \\ &= 2\arcsin(\sqrt{x}) + C \end{align*}

Answer 2

Hint

With the substitution $x= \sin^2(u),$ we have $dx = 2 \sin(u) \cos(u) \ du$ Figure out the transformed integrand.