Question

The value of $\int \frac{dx}{(a^2+x^2{)}^{\frac{3}{2}}}$ is

• A. $\frac{x}{a^2(x^2+a^2{)}^{\frac{1}{2}}}+C$
• B. $\frac{x(x^2+a^2{)}^{\frac{1}{2}}}{a^2}+C$
• C. $\frac{a^{2}x}{(x^2+a^2{)}^{\frac{1}{2}}}+C$
• D. $a^{2}x(x^2+a^2{)}^{\frac{1}{2}}+C$

Answer 1

The correct option is A $\frac{x}{a^2(x^2+a^2{)}^{\frac{1}{2}}}+C$

$I=\int \frac{dx}{(a^2+x^2{)}^{\frac{3}{2}}}$
Substituting $x=a\tan \theta$
$\Rightarrow dx=a{\sec }^2\theta d\theta \Rightarrow I=\int \frac{a{\sec }^2\theta d\theta }{a^3{\sec }^3\theta }=\frac{1}{a^2}\int \cos \theta d\theta =\left(\frac{1}{a^2}\right)\sin \theta +C,\sin \theta =\frac{x}{\sqrt{(x^2+a^2)}}=\frac{x}{a^2\sqrt{(x^2+a^2)}}+C$