Question

Solve :
$\int \frac{1+x\cos x}{x(1-x^2{e}^{2\sin x})}dx=k\ell n\sqrt{\frac{x^2{e}^{2\sin x}}{1-x^2{e}^{2\sin x}}}+C$ then $k$ is equal to :

Answer 1

$\begin{array}{c}\int \frac{1+x\cos x}{x\left(1-x^2{e}^{2\sin x}\right)}dx\\ putx{e}^{\sin x}=t\\ e^{\sin x}\left(1+x\cos x\right)dx=dt\\ =\int \frac{dt}{t\left(1-t^2\right)}\\ =\log t-\frac{1}{2}\log \left(1-t\right)-\frac{1}{2}\log \left(t+1\right)+C\\ =\frac{1}{2}\left(\log \left(\frac{t^2}{1+t^2}\right)\right)+C\\ =\frac{1}{2}\log \left(\frac{x^2{e}^{2\sin x}}{1-x^2{e}^{2\sin x}}\right)+C\\ \therefore K=1\end{array}$