1. Hyperbolic functions

Evaluate:

$\displaystyle \int\tanh 2x\, ln(cosh 2x)\, dx$

$\displaystyle \int\ x^2, csch^2\, x^3\, dx$

$\displaystyle \int\ x cosh x^2\, sinh x^2\, dx$

2. $\displaystyle \int \tanh(2x)\ln(\cosh(2x))dx$

let u = $\displaystyle \ln(\cosh(2x))$

du = $\displaystyle \frac{1}{\sinh(2x)}.2\cosh(2x)$ <<< using chain rule derivative

substitute du and cancellation
$\displaystyle \int \frac{\sinh{2x}}{\cosh{2x}}u\frac{\cosh{2x}}{2\sin h{2x}}du$

take 1/2 out
$\displaystyle \frac{1}{2}\int u du$

integrate
$\displaystyle \frac{1}{2} (\frac{1}{2}u^2)$
substitute u = $\displaystyle \ln(\cosh(2x))$

$\displaystyle \frac{1}{4} \ln^2(\cosh{2x}) + C$

3. $\displaystyle \int x^2 csch^2{x^3} dx$

let u = x^3
du = 3x^2

$\displaystyle \int csch^2{u} \frac{du}{3}$

integrate and
substitute u
$\displaystyle -\frac{1}{3}\coth{x^3} + C$

4. $\displaystyle \int x \cosh x^2, \sinh x^2 dx$

let u = x^2
du = 2xdx
$\displaystyle \int \sinh{u} \cosh{u} \frac{du}{2}$

let y = sinh{u}
dy = cosh{u}

$\displaystyle \frac{1}{2}\int y dy$

$\displaystyle \frac{1}{2} (\frac{y^2}{2})$

from y = sinh{u}

$\displaystyle \frac{1}{4} \sinh^2{u} + C$

$\displaystyle \frac{1}{4} \sinh^2{x^2} + C$