Question

Solve:- $\int \frac{5x+3}{\sqrt{x^2+4x+10}}dx$

Answer 1

$\int \frac{5x+3}{\sqrt{x^2+4x+10}}dx$

$5x+3=A(2x+4)+B$

$A=\frac{5}{2},3=4A+B$

$B=-7$

$=\int \frac{\frac{5}{2}(2x+4)}{\sqrt{x^2+4x+10}}dx+\int \frac{-7}{\sqrt{(x+2{)}^2+6}}$

$=\int \frac{5}{2}\times \frac{1}{\sqrt{t}}-7\int \frac{1dx}{(x+2{)}^2+(\sqrt{6}{)}^2}$

$=\frac{5}{2}\times 2\sqrt{t}-7\times \log \left|\right(x+2)+\sqrt{x^2+4x+10}|+C$

$=5\sqrt{x^2+4x+10}-7\log \left|\right(x+2)+\sqrt{x^2+4x+10}|+C$