Question

$\int \frac{dx}{x^2\sqrt{x^2+a^2}}=$

• A. $-\frac{\sqrt{x^2+a^2}}{a^{2}x}+c$
• B. $\frac{\sqrt{x^2+a^2}}{a^{2}x}+c$
• C. $-\frac{a^{2}x}{\sqrt{x^2+a^2}}+c$
• D. $\frac{a^{2}x}{\sqrt{x^2+a^2}}+c$

Answer 1

The correct option is A $-\frac{\sqrt{x^2+a^2}}{a^{2}x}+c$

$I=\int \frac{dx}{x^2\sqrt{x^2+a^2}}\left\{\begin{array}{c}Letx=a\tan \theta \\ \theta ={\tan }^{-1}\left(\frac{x}{a}\right)\\ dx=a{\sec }^2\theta d\theta \end{array}\right\}$

$I=\int \frac{a{\sec }^2\theta d\theta }{a^2{\tan }^2\theta \sqrt{a^2(1+{\tan }^2\theta )}}$

$I=\int \frac{{\sec }^{}\theta d\theta }{a^2{\tan }^2\theta }$

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

$I=\frac{1}{a^2}\int \frac{\cos \theta }{{\sin }^2\theta }d\theta$
$I=\frac{1}{a^2}\int \csc \theta \cot \theta d\theta$

$I=-\frac{\csc \theta }{a^2}+c$

$=-\frac{\sqrt{x^2+a^2}}{a^{2}x}+c$