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