Question

Find $\int \frac{2cosx}{(1-sinx)(1+si{n}^{2}x)}dx$

Answer 1

Let I = $\int \frac{2cosx}{(1-sinx)(1+si{n}^{2}x)}dx$ [Put $sinx=t\Rightarrow cosxdx=dt$]

$\therefore I=\int \frac{2dt}{(1-t)(1+t^2)}$....(i)

Consider $\frac{2}{(1-t)(1+t^2)}=\frac{A}{1-t}+\frac{B\left(2t\right)}{1+t^2}+\frac{C}{1+t^2}\Rightarrow 2=A(1+t^2)+B\left(2t\right)(1-t)+C(1-t)$

On comparing the coefficients of like terms , we get $A=1,B=\frac{1}{2},C=1$

By(i), $I=\int \frac{dt}{1-t}+\frac{1}{2}\int \frac{2t}{1+t^2}dt+\int \frac{1}{1+t^2}dt=-log|1-t|+\frac{1}{2}log|1+t^2|+ta{n}^{-1}t+c\therefore I=\frac{1}{2}log|1+si{n}^{2}x|-log|1-sinx|+ta{n}^{-1}\left(sinx\right)+c$