Question

Calculus help please! Click the link to see the integral.

$\int\limits { \frac{cosx}{ \sqrt{sinx} } } \, dx$

Answer 1

Let $u=\sin x$, then $\mathrm du=\cos x\,\mathrm dx$. Now

$\displaystyle\int\frac{\cos x}{\sqrt{\sin x}}\,\mathrm dx=\int\frac{\mathrm du}{\sqrt u}=\int u^{-1/2}\,\mathrm du$

The power rule for integration gives

$\displaystyle\int u^{-1/2}\,\mathrm du=2u^{1/2}+C=2\sqrt{\sin x}+C$