Question

Evaluate $\int \frac{x}{\sqrt{x^2+2}}dx$

• A. $I=\sqrt{x^2-2}+C$
• B. $I=\sqrt{x^2+2}+C$
• C. $I=\sqrt{x^3+2}+C$
• D. $I=\sqrt{x^3-2}+C$

Answer 1

Answer: (B)

$I=\sqrt{x^2+2}+C$

Consider the given integral.

$I=\int \frac{x}{\sqrt{x^2+2}}dx$

Let $t=x^2+2$

$\frac{dt}{dx}=2x+0$

$\frac{dt}{2}=xdx$

Therefore,

$I=\frac{1}{2}\int \frac{dt}{\sqrt{t}}$

$I=\frac{1}{2}\left(2\sqrt{t}\right)+C$

$I=\sqrt{t}+C$

On putting the value of $t$, we get

$I=\sqrt{x^2+2}+C$

Hence, this is the answer.