Question

Integrate following with respect to $x$.
(i) $\sqrt{x^2+4}$ (ii) $\sqrt{x^2+2x+5}$

Answer 1

(i) $\int \sqrt{x^2+4}dx$

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

$=\frac{x}{2}\sqrt{x^2+4}+\frac{4}{2}si{n}^{-1}\left(\frac{x}{2}\right)+C..........[\int \sqrt{x^2+a^2}dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}si{n}^{-1}\left(\frac{x}{a}\right)+C]$

$\therefore \int \sqrt{x^2+4}dx=\frac{x}{2}\sqrt{x^2+4}+2si{n}^{-1}\left(\frac{x}{2}\right)+C$

(ii) $\int \sqrt{x^2+2x+5}dx$

$=\int \sqrt{(x+1{)}^2+2^2}dx$

$=\frac{(x+2)}{2}\sqrt{x^2+2x+5}+\frac{4}{2}si{n}^{-1}\left(\frac{x+1}{2}\right)+C$

$\therefore \int \sqrt{x^2+2x+5}dx=\left(\frac{x+1}{2}\right)\sqrt{x^2+2x+5}+2si{n}^{-1}\left(\frac{x+1}{2}\right)+C$