Find: $\int_0^{\infty}\frac{\sinh x}{1+\cosh^2x}dx$

Question

$$\int_0^{\infty}\frac{\sinh x}{1+\cosh^2x}dx$$

Here's what I've attempted:

Using the identity $1+\cosh^2x=\sinh^2x$ I got:

$$\int_0^{\infty}\frac{\sinh x}{\sinh^2x}dx=\int_0^{\infty}\frac1{\sinh x}dx$$

Is this right? However, where can I go from here?

Answer 1

The identity you are using is not correct. The correct one is

\begin{equation} \cosh^2 x -\sinh^2 x = 1 \end{equation}

Moreover this integration is a simple change of variables:

\begin{equation} \begin{aligned} &\int_0^{+\infty} \frac{\sinh x}{1 + \cosh^2 x} dx\\ =& \int_0^{+\infty} \frac{1}{1 + \cosh^2 x} d(\cosh x)\\ =& \int_1^{+\infty} \frac{1}{1+y^2} dy\\[0.5em] =&~ \tan^{-1} y~|_{+\infty} - \tan^{-1} y~|_{1} \\[0.5em] =&~ \frac{\pi}{2} - \frac{\pi}{4}\\[0.5em] =&~ \frac{\pi}{4} \end{aligned} \end{equation}

Answer 2

Set $x=\text{arccosh(y)}$, $dx=\frac{1}{\sqrt{y^2-1}}dy$ ,

furthermore we know that $\sinh(\text{arccosh(y)})=\sqrt{y^2-1}$ .

Note that this part exactly cancels the stuff from the differential and therefore we get: $$ I=\int_1^{\infty}dy\frac{1}{1+y^2}=\text{arctan}(\infty)-\text{arctan}(1)=\frac{\pi}{4} $$