Question

Evaluate $\int \frac{2x+5}{\sqrt{x^2-2x+10}}dx$.

Answer 1

$\int \frac{2x+5}{x^2-2x+10}=\int \frac{2(x-1)+7}{\sqrt{(x-1{)}^2+9}}dx$

$=\int \frac{2(x-1)dx}{\sqrt{(x-1{)}^2+9}}+7\int \frac{dx}{\sqrt{(x-1{)}^2+3^2}}$

$=\int \left[\right(x-1{)}^2+9{]}^{1/2}d\left[\right(x-1{)}^2+9]+7\int \frac{dx}{\sqrt{(x-1{)}^2+3^2}}$

$=\frac{2}{3}\left[\right(x-1{)}^2+9{]}^{3/2}+7\ln |(x-1)+\sqrt{(x-1{)}^2+9}|+C$

$=\frac{2}{3}(x^2-2x+10{)}^{3/2}+7\ln |x-1+\sqrt{x^2-2x+10}|+C$