Question

The value of $\int \frac{x \, d x}{\left(\right. x + 3 \left.\right) \sqrt{x + 1}}$ is (where, $c$ is the constant of integration)

• A. $2\sqrt{x + 1}+3 \, \, tan^{- 1}\sqrt{x + 1}+c$
• B. $2\sqrt{x + 1}+3\sqrt{2} \, \, tan^{- 1}\sqrt{\frac{x + 1}{2}}+c$
• C. $2\sqrt{x + 1}-3\sqrt{2} \, \, tan^{- 1}\sqrt{\frac{x + 1}{2}}+c$
• D. $2\sqrt{x + 1}-3 \, \, tan^{- 1}\sqrt{x + 1}+c$

Answer 1

Answer: (C)

Substitute $\sqrt{x + 1}=t$ , we get,
$2\int \frac{t^{2} - 1}{t^{2} + 2} \, \, dt=2t-6\int \frac{1}{t^{2} + 2} \, dt$
$=2t-3\sqrt{2}tan^{- 1}\frac{t}{\sqrt{2}}+c$
$=2\sqrt{x + 1}-3\sqrt{2}tan^{- 1}\sqrt{\frac{x + 1}{2}}+c$