Question

Evaluate

$\int$ $\frac{3x+3}{\sqrt{x^2+5x+6}}$

Answer 1

$\int \frac{3x+3}{\sqrt{x^2+5x+6}}dx$

Let $t^2=x^2+5x+6$

$\Rightarrow 2tdt=(2x+5)dx$

Substituting,

$=\frac{3}{2}\int \frac{2x+5-3}{\sqrt{x^2+5x+6}}dx$

$=\frac{3}{2}\int \frac{2x+5}{\sqrt{x^2+5x+6}}dx-\frac{9}{2}\int \frac{1}{\sqrt{x^2+5x+6}}dx$

$=\frac{3}{2}\int \frac{2tdt}{t}-\frac{9}{2}\int \frac{1}{\sqrt{x^2.2.\frac{1}{2}.5.x+\frac{25}{4}-\frac{25}{4}+6}}dx$

$=\frac{3}{2}\int \frac{2tdt}{t}-\frac{9}{2}\int \frac{1}{\sqrt{{(x+\frac{5}{2})}^2-{\left(\frac{1}{2}\right)}^2}}dx$

$=\frac{3}{2}\times 2\int dt-\frac{9}{2}\int \frac{1}{\sqrt{{(x+\frac{5}{2})}^2-{\left(1/2\right)}^2}}dx$

$=3\int dt-\frac{9}{2}\int \frac{1}{\sqrt{{(x+\frac{5}{2})}^2-{\left(1/2\right)}^2}}dx$

$=3\sqrt{x^2+5x+6}-\frac{9}{2}log(x+\frac{5}{2}+\sqrt{{(x+\frac{5}{2})}^2-\frac{1}{4}})+C$