Question

Evaluate:
$\int \frac{2x+3}{\sqrt{4x+3}}dx=$

Answer 1

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

$\Rightarrow \frac{1}{2}\int \frac{(4x+3)}{\sqrt{4x+3}}+\frac{3}{2}\int \frac{dx}{\sqrt{4x+3}}$

$\Rightarrow 4x+3=m$

$dx=\frac{dm}{4}$

$\Rightarrow \frac{1}{8}\int \frac{m}{\sqrt{m}}dm+\frac{3}{8}\int \frac{dm}{\sqrt{m}}$

$\Rightarrow \frac{1}{8}\int \sqrt{m}dm+\frac{3}{8}\int {\left(\sqrt{m}\right)}^{-1}dm$

$\Rightarrow \frac{1}{8}\times \frac{2}{3}\times m^{3/2}+\frac{3}{8}\times 2\times \sqrt{m}+c$

$\Rightarrow \frac{1}{12}{(4x+3)}^{3/2}+\frac{3}{4}\sqrt{4x+3}+c$