Question

Integrate the function: $\frac{4x+1}{\sqrt{2x^2+x-3}}$

Answer 1

To find $\int \frac{4x+1}{\sqrt{2x^2+x-3}}dx$

Let $2x^2+x-3=t^2$
Differentiating both sides w.r.t $x$

$4x+1=2t\frac{dt}{dx}\Rightarrow (4x+1)dx=2tdt$

Now,

$\int \frac{4x+1}{\sqrt{2x^2+x-3}}dx$

$=\int \frac{2tdt}{t}$

$=\int 2.dt$

$=2\int 1.dt$

$=2t+C$

$=2\sqrt{2x^2+x-3}+C$

where $C$ is a constant of integration