Question

$\int \sqrt{x^2+2x+5}dx$ is equal to
$($where $C$ is integration constant$)$

• A. $\log |x+\sqrt{x^2+2x+5}|+C$
• B. $\frac{1}{2}{\tan }^{-1}\left[\frac{x+1}{2}\right]+C$
• C. $\frac{1}{2}(x+1)\sqrt{x^2+2x+5}+2\log |x+1+\sqrt{x^2+2x+5}|+C$
• D. $\frac{1}{2}(x+1)\sqrt{x^2+2x+5}+2{\sin }^{-1}\left(\frac{x+1}{2}\right)+C$

Answer 1

The correct option is C $\frac{1}{2}(x+1)\sqrt{x^2+2x+5}+2\log |x+1+\sqrt{x^2+2x+5}|+C$

$I=\int \sqrt{x^2+2x+5}dx=\int \sqrt{(x+1{)}^2+4}dx$
Put $x+1=y\Rightarrow dx=dy$
$I=\int \sqrt{y^2+2^2}dy=\frac{y}{2}\sqrt{y^2+4}+\frac{4}{2}\log |y+\sqrt{y^2+4}|+C=\frac{1}{2}(x+1)\sqrt{x^2+2x+5}+2\log |x+1+\sqrt{x^2+2x+5}|+C[\because \int \sqrt{x^2+a^2}dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\log |x+\sqrt{x^2+a^2}|+C]$