Integration with multiple constants

Question

Question:

$$\int \frac{ax^2-b}{x\sqrt{{c^2x^2-(ax^2+b)^2}}}\ \text dx$$

My approach:

I can't understand whether I should integrate it normally or use a trigonometric function.

Answer 1

Let $$I =\int\frac{(ax^2-b)}{x\sqrt{c^2x^2-(ax^2+b)^2}}dx = \int\frac{ax^2-b}{x\cdot x \sqrt{c^2-(ax+\frac{b}{x})^2}}dx$$

So $$I = \int\frac{a-\frac{b}{x^2}}{\sqrt{c^2-(ax+\frac{b}{x})^2}}dx$$

Put $\displaystyle \left(ax+\frac{b}{x}\right) = t\;,$ Then $\displaystyle \left(a-\frac{b}{x^2}\right)dx = dt$

$$I = \int\frac{1}{\sqrt{c^2-t^2}}dt = \arcsin\left(\frac{t}{c}\right)+\mathcal{C}$$

So $$I =\arcsin\bigg(\frac{ax^2+b}{cx}\bigg)+\mathcal{C}$$