Question

Evaluate $\int {\sin }^{\frac{3}{4}}x\cos xdx$.

Answer 1

$\int {\sin }^{\frac{3}{4}}x\cos xdx$

Put $t=\sin x\Rightarrow dt=\cos xdx$

$=\int t^{\frac{3}{4}}dt$

$=\frac{t^{\frac{3}{4}+1}}{\frac{3}{4}+1}+c$

$=\frac{t^{\frac{3+4}{4}}}{\frac{3+4}{4}}+c$

$=\frac{t^{\frac{7}{4}}}{\frac{7}{4}}+c$

$=\frac{4}{7}{t}^{\frac{7}{4}}+c$ , where $c$ is the constant of integration

$=\frac{4}{7}{\sin }^{\frac{7}{4}}x+c$, where $t=\sin x$