Evaluate the following definite integrals: _ 2 ^3 x x^2+1 dx - Vidyadip

Question

Evaluate the following definite integrals:
$\int_{2}^\limits{3}\frac{\text{x}}{\text{x}^2+1} \text{ dx}$

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Answer

Let $\text{I}=\int_{2}^\limits{3}\frac{\text{x}}{\text{x}^2+1} \text{ dx}$ Then,
$\text{I}=\frac{1}{2}\int_{2}^\limits{3}\frac{2\text{x}}{\text{x}^2+1} $
$\Rightarrow\text{I}=\frac{1}{2}\big[\log(\text{x}^2-1)\big]^3_2$
$\Rightarrow\text{I}=\frac{1}{2}\big(\log10-\log5\big)$
$\Rightarrow\text{I}=\frac{1}{2}\log\frac{10}{5}$ $\Big[\because\log\text{a}-\log\text{b}=\log\frac{\text{a}}{\text{b}}\Big]$
$\Rightarrow\text{I}=\frac{1}{2}\log2$

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