Ingerals, what should I substitute.

Question

What should I substitute to get from :

$\int \frac{1-x^2}{x \cdot \sqrt[]{1-x^2} }dx$

integral which will be:

$\int \frac{a_nx^n+...+a_1x+a_0}{\sqrt[]{1-x^2} }dx$

Answer 1

Let $$I = \int\frac{(1-x^2)}{x\sqrt{1-x^2}}dx = \int\frac{\sqrt{1-x^2}}{x}dx$$

Put $x= \sin \theta$ Then $dx = \cos \theta d\theta$

$$I = \int\frac{\cos \theta \cdot \cos \theta}{\sin \theta}d\theta = \int \frac{1-\sin^2 \theta}{\sin \theta}d\theta$$

Answer 2

Let $t^2=1-x^2$. Then is $t=\sqrt{1-x^2}$, and $x^2=1-t^2$ or $x=\sqrt{1-t^2}$ so $dx = \frac{-t}{\sqrt{1-t^2}}dt$. Now convert the integral with $x$ to an integral with $t$: $I = \int\frac{t^2\cdot(-t)}{\sqrt{1-t^2}\cdot t \cdot \sqrt{1-t^2}}dt = -\int\frac{t^3}{(1-t^2)\cdot t}dt$. You should be able to solve this now.