Evaluate the following : _1^2 5x^2 x^2+4x+3 dx - Vidyadip

Question

Evaluate the following :
$\int\limits_1^2\frac{5\text{x}^2}{\text{x}^2+4\text{x}+3}\ \text{dx}$

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Answer

$\text{I}=\int\limits_1^2\frac{5\text{x}^2}{\text{x}^2+4\text{x}+3}\ \text{dx}=5\int\limits_1^21-\frac{4\text{x}+3}{\text{x}^2+4\text{x}+3}\text{ }\text{dx} $
$=5\big[\text{x}\big]_1^2-10\int\limits_1^2\frac{2\text{x}+4-\frac{5}{2}}{\text{x}^2+4\text{x}+3}\ \text{dx}$
$=5-10\big[\log|\text{x}^2+4\text{x}+3|\big]_1^2+25\int\limits_1^2\frac{1}{(\text{x}+2)^2-(1)^2}\ \text{dx}$
$=5-10\ \log\ \frac{15}{8}+25\ .\ \frac{1}{2}\bigg[\log\begin{vmatrix}\frac{\text{x}+2-1}{\text{x}+2+1}\end{vmatrix}\bigg]_1^2$
$=5-10\ \log\frac{15}{8}+\frac{25}{2}\ \log\frac{6}{5}$

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