Question

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

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

Answer 1

Answer: (C)

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