Question

$\displaystyle\int_{0}^{\infty}\frac{dx}{(x+\sqrt{x^2+1})^3 }=$

• A. $\frac{3}{8}$
• B. $\frac{1}{8}$
• C. $-\frac{3}{8}$
• D. $-\frac{1}{8}$

Answer 1

Answer: (A)

Put $x = tan \, \theta \therefore dx=sec^{2} \, \theta\,d\theta$
When $x = 0, \theta=0, x=\infty, \theta=\pi /2$
$\therefore $ given integral
$=\int\limits_{0}^{\pi /2} \frac{cos\, \theta d\theta}{\left(1+sin\,\theta\right)^{3}}=\left|\frac{\left(1+sin\,\theta\right)^{-2}}{-2}\right|_{0}^{\pi /2}$
$=-\frac{1}{8}+\frac{1}{2}=\frac{3}{8}$