Evaluate the integral $\int \frac{1 + x\cos(x)}{x(1-x^2e^{2\sin(x)})}\, dx$

Question

I'm trying to evaluate the following integral: $$\int \frac{1 + x\cos(x)}{x(1-x^2e^{2\sin(x)})}\, dx$$ I have no idea how to approach this problem at all. Could someone please guide me through the steps to solve this integral? Are there any specific trigonometric identities or techniques that I should apply? Any insights or hints would be greatly appreciated.

Thank you in advance for your assistance!

Answer 1

Given,

$$I=\int \frac{1 + x\cos(x)}{x(1-x^2e^{2\sin(x)})}\, dx$$

Substitue $xe^{sinx}=t$

$$I=(-1)\int \frac{1}{t^3-t} \,dt$$

For generalizing;

$$I=\int \frac{1}{t(t^n-1)} \,dt=\frac{ln(t^n-1)}{n}-ln(t)+c$$

$$Or$$

$$I=\frac{ln\left(1-\frac{1}{t^n}\right)}{n}+c$$

Put $n=2$ in the above generalized integral;

$$I=(-1)\frac{ln\left(1-\frac{1}{t^2}\right)}{2}+c$$

I think you can take it from here.

Extras -

$xe^{sinx}=t$

$e^{sinx}(1+xcosx)dx=dt$

$\frac{e^{sinx}(1+xcosx)}{x}dx=\frac{1}{x}dt$

$\frac{(1+xcosx)}{x}dx=\frac{1}{t}dt$