Question

Solve: $\int \frac{{\cos }^{3}x}{\sqrt{\sin x}}dx$.

Answer 1

Now,

$\int \frac{{\cos }^{3}x}{\sqrt{\sin x}}dx$

$=\int \frac{(1-{\sin }^{2}x).\cos x}{\sqrt{\sin x}}dx$

Let $\sin x=z$ or, $\cos xdx=dz$.

Using these in above expression we get,

$=\int \frac{(1-z^2)dz}{\sqrt{z}}$

$=\int \frac{1}{\sqrt{z}}dz$$-\int z^{\frac{3}{2}}dz$

$=2\sqrt{z}-\frac{2}{5}{z}^{\frac{5}{2}}+c$ [ Where $c$ is constant]

$=2\sqrt{\sin x}-\frac{2}{5}(\sin x{)}^{\frac{5}{2}}+c$