Question

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

Answer 1

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