Evaluation of $k$ for given integral

Question

If $$\int \frac{1+x \cos x}{x(1-x^2 e^{2 \sin x})}dx=k \ln \sqrt{\frac{x^2 e^{2 \sin x}}{1-x^2 e^{2 \sin x}}}+C$$

Is there any method to find $k$ without actually integrating L.H.S. because integration is taking long steps.

Answer 1

When you do use integration, substitute $u=\frac{1}{x^2e^{2\sin(x)}-1}$ and $\text{d}u=-\frac{2xe^{2\sin(x)}\left(1+x\cos(x)\right)}{\left(x^2e^{2\sin(x)}-1\right)^2}\space\text{d}x$:

$$\int\frac{1+x\cos(x)}{x\left(1-x^2e^{2\sin(x)}\right)}\space\text{d}x=-\frac{1}{2}\int\frac{1}{1+u}\space\text{d}u=\text{C}-\frac{\ln\left|1+u\right|}{2}=\text{C}-\frac{\ln\left|1+\frac{1}{x^2e^{2\sin(x)-1}}\right|}{2}$$