Evaluate the following : x+2 ((x-2)(x-3) dx - Vidyadip

Question

Evaluate the following :
$\int\frac{\text{x+2}}{\sqrt{(\text{(x-2)(x-3)}}}\text{dx}$

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Answer

$\text{Here I}=\int\frac{\text{x+2}}{\text{x}^2-5\text{x}+6}\text{ dx}=\frac{1}{2}\int\frac{2\text{x}-5+9}{\sqrt{{\text{x}^2}-5\text{x}+6}}\text{dx} $
$=\frac{1}{2}\int\frac{2\text{x}-5}{\sqrt{\text{x}^2-5\text{x}+6}}\ \text{dx}+\frac{9}{2}\int\frac{1}{\sqrt{\bigg(\text{x}-{5}/{2}\bigg)^2-\bigg(\frac{1}{2}\bigg)^2}}\ \text{dx}$
$\sqrt{\text{x}^2+4\text{x}+3}+\frac{9}{2}\log\begin{vmatrix}\bigg(\text{x}-\frac{5}{2}\bigg)+\sqrt{\text{x}^2-5\text{x}+6}\end{vmatrix}+\text{c}$

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