Question

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

Answer 1

$$\begin{gathered} \int \frac{{\cos }^{3}x}{\sqrt{\sin x}}dx \\ =\int \frac{{\cos }^{2}x·\cos x}{\sqrt{\sin x}}dx \\ =\int \frac{\left(1-{\sin }^{2}x\right)\cos x}{\sqrt{\sin x}}dx \\ \mathrm{Let}\sin x=t \\ \Rightarrow \cos x=\frac{dt}{dx} \\ \Rightarrow \cos xdx=dt \\ \mathrm{Now},\int \frac{\left(1-{\sin }^{2}x\right)\cos x}{\sqrt{\sin x}}dx \\ =\int \frac{\left(1-t^2\right)}{\sqrt{t}}·dt \\ =\int \left(\frac{1}{\sqrt{t}}-t^{\frac{3}{2}}\right)dt \\ =\int \left(t^{-\frac{1}{2}}-t^{\frac{3}{2}}\right)dt \\ =\left[\frac{t^{-{\displaystyle \frac{1}{2}}+1}}{{\displaystyle -\frac{1}{2}}+1}-\frac{t^{{\displaystyle \frac{3}{2}}+1}}{{\displaystyle \frac{3}{2}}+1}\right]+\mathrm{C} \\ =2\sqrt{t}-\frac{2}{5}{t}^{\frac{5}{2}}+\mathrm{C} \\ =2\sqrt{\sin x}-\frac{2}{5}{\sin }^{\frac{5}{2}}x+\mathrm{C} \end{gathered}$$