1. ## Integral 01

$\displaystyle \int \frac{dx}{x^4\sqrt{1+x^2}}$

$\displaystyle \frac{(2x^2-1)(1+x^2)\frac{1}{2}}{3x^3}$

2. $\displaystyle \int \frac{dx}{x^4\sqrt{1+x^2}}$

You can use trigonometric subtitution.

Let:
$\displaystyle x = \tan{\theta}$

$\displaystyle dx = \sec^2{\theta} d\theta$

$\displaystyle \int \frac{\sec^2{\theta}}{(\tan{\theta})^4\sqrt{1+\tan ^2{\theta}}} d\theta$

$\displaystyle \int \frac{\sec^2{\theta}}{(\tan{\theta})^4\sec{\theta} } d\theta$

$\displaystyle \int \frac{\sec{\theta}}{(\tan{\theta})^4} d\theta$

$\displaystyle \int \frac{1}{\cos{\theta}} \cdot \frac{cos^4{\theta}}{\sin^4{\theta}} d\theta$

$\displaystyle \int \frac{cos^3{\theta}}{\sin^4{\theta}} d\theta$

$\displaystyle \int \frac{(1-sin^2{\theta})\cos{\theta}}{\sin^4{\theta}} d\theta$

Let
$\displaystyle u = \sin{\theta}$

$\displaystyle du = \cos{\theta} d\theta$

$\displaystyle \int \frac{1-u^2}{u^4} du$

Can you continue?

3. Think in a reciprocal substitution, like $\displaystyle u=\frac1x,$ the integral equals $\displaystyle -\int{\frac{u^{3}}{\sqrt{1+u^{2}}}\,du}.$ Now put $\displaystyle z^2=1+u^2.$

4. Look my solution

$\displaystyle \int \frac{dx}{x^4\sqrt{1+x^2}} = \int x^(-4) (1+x^2)^-\frac{1}{2}dx$

$\displaystyle 1+x^2=z^2$
$\displaystyle x^2=z^2-1$
$\displaystyle dx= \frac{zdz}{\sqrt{z^2-1}}$

$\displaystyle \int (z^2+1)^(-2) (z^2)^\frac{1}{2}\frac{z}{\sqrt{z^2-1}}$