Question

Evaluate: $\int \sqrt{5-2x+x^2}dx$

Answer 1

$\int \sqrt{5-2x+x^2}dx$
$=\int \sqrt{x^2-2x+1+4}dx$
$=\int \sqrt{{(x-1)}^2+2^2}dx$
$=\frac{x-1}{2}\sqrt{x^2-2x+5}+2\log |(x-1)+\sqrt{x^2-2x+5}+c|$

$(\because \int \sqrt{x^2+a^2}dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\log |x+\sqrt{x^2+a^2}|)$

Where $c$ is constant of integration.
Hence, the required integration is $\frac{x-1}{2}\sqrt{x^2-2x+5}+2\log |(x-1)+\sqrt{x^2-2x+5}+c|$