Question

Evaluate the following integrals.
$\int \sqrt{1+3x-x^2}dx$

Answer 1

$I=\int \sqrt{1+3x-x^2}dx$

$=\int \sqrt{1-[{(x-\frac{3}{2})}^2-\frac{9}{4}]}dx$

$=\int \sqrt{\frac{10}{4}-{(x-\frac{3}{2})}^2}dx$

$=\int \sqrt{{\left(\frac{\sqrt{13}}{2}\right)}^2-{(x-\frac{3}{2})}^2}dx$

$=\frac{x-\frac{3}{2}}{2}\sqrt{{\left(\frac{\sqrt{13}}{2}\right)}^2-{(x-\frac{3}{2})}^2}+\frac{\frac{(\sqrt{13}}{2}{)}^2}{2}{\sin }^{-1}\frac{x-\frac{3}{2}}{\frac{\sqrt{13}}{2}}+C$

$\frac{2x-3}{4}\sqrt{\frac{13}{4}-{(x-\frac{3}{2})}^2}+\frac{13}{8}{\sin }^{-1}\frac{2x}{\sqrt{13}}+C$