Question

Evaluate: $\int \frac{(a{x}^2-b)dx}{x\sqrt{c^2{x}^2-(a{x}^2+b{)}^2}}$

• A. ${\sin }^{-1}\left(\frac{b{x}^2-a}{cx}\right)+k$
• B. ${\sin }^{-1}\left(\frac{b{x}^2+a}{cx}\right)+k$
• C. ${\sin }^{-1}\left(\frac{a{x}^2-b}{cx}\right)+k$
• D. ${\sin }^{-1}\left(\frac{a{x}^2+b}{cx}\right)+k$

Answer 1

The correct option is D ${\sin }^{-1}\left(\frac{a{x}^2+b}{cx}\right)+k$

$I=\int \frac{a{x}^2-b}{x\sqrt{c^2{x}^2-(a{x}^2+b^2)}}dx=\int \frac{a-\frac{b}{x^2}}{\sqrt{c^2-{(ax+\frac{b}{x})}^2}}dx$
Put $ax+\frac{b}{x}=t\Rightarrow (a-\frac{b}{x^2})dx=dt$
$I=\int \frac{dt}{\sqrt{c^2-t^2}}={\sin }^{-1}\frac{t}{c}={\sin }^{-1}\frac{(ax+\frac{b}{x})}{c}$