Question

Solve : $\int \frac{\sin x{\cos }^{3}x}{1+{\cos }^{2}x}dx$

Answer 1

$\int \frac{\sin x{\cos }^{3}x}{1+{\cos }^{2}x}dx$

$\cos x=p$

$-sinxdx=dp$

$\Rightarrow -\int \frac{p^3}{1+p^2}dp$

$1+p^2=k\Rightarrow p^2=(k-1)$

$2pdp=dk$

$\Rightarrow \int \frac{(k-1)dk}{2\left(k\right)}$

$\therefore \int \frac{1}{2}dx+\frac{1}{2}\int \frac{dx}{k}$

$\therefore \int -\frac{k}{2}+\frac{1}{2}ln|k|$

$\therefore \frac{-(1+p^2)}{2}+\frac{1}{2}ln|1+p^2|$

$\therefore \frac{1}{2}[ln|1+{\cos }^{2}x)-(1+{\cos }^{2}x)]$