Question

$\int_{}^{}\frac{dx}{(a^{2} + x^{2})^{3/2}}$ is equal to

• A. $\frac{x}{(a^{2} + x^{2})^{1/2}}$ + c
• B. $\frac{x}{a^{2}(a^{2} + x^{2})^{1/2}}$+ c
• C. $\frac{1}{a^{2}(a^{2} + x^{2})^{1/2}}$ + c
• D. None of these

Answer 1

Answer: (B)

$\frac{x}{a^{2}(a^{2} + x^{2})^{1/2}}$+ c

Answer 2

$I = \int_{}^{}\frac{dx}{(a^{2} + x^{2})^{3/2}}$

Put $x = a\tan\theta$ $\Rightarrow dx = a\sec^{2}\theta d\theta$

$\therefore I = \int_{}^{}\frac{a\sec^{2}\theta d\theta}{(a^{2} + a^{2}\tan^{2}\theta)^{3/2}}$ $= \int_{}^{}{\frac{a\sec^{2}\theta d\theta}{a^{3}(\sec^{2}\theta)^{3/2}} = \frac{1}{a^{2}}\int_{}^{}\frac{\sec^{2}\theta d\theta}{\sec^{3}\theta}}$ $\Rightarrow I = \frac{1}{a^{2}}\int_{}^{}{\frac{d\theta}{\sec\theta} = \frac{1}{a^{2}}\int_{}^{}{\cos\theta ⥂ d\theta}}$⇒ $I = \frac{1}{a^{2}}\sin\theta + c$

$\Rightarrow$ $\frac{x}{a^{2}(x^{2} + a^{2})^{1/2}}$ + c