Evaluate the integral in Exercise: _ 0 ^ 1 x x^ 2 +1 dx - Vidyadip

Question

Evaluate the integral in Exercise:
$\int_{0}^{1}\frac{\text{x}}{\text{x}^{2}+1}\text{dx}$

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Answer

$\text{Let}\ \text{I}=\int\limits_{0}^{1}\frac{\text{x}}{\text{x}^{2}+1}\text{dx}$
$\text{put}\ \text{x}^{2}+1=\text{y},\ \ \therefore 2\text{x}\ \text{dx}=\text{dy},\ \text{or}\ \text{x}\ \text{dx}=\frac{1}{2}\text{dy}$
$\text{when}\ \text{x}=2,\ \text{y}=5$
$\text{when}\ \text{x}=3,\ \text{y}=10$
$\therefore |=\frac{1}{2}\int^{10}_{5}\frac{\text{dy}}{\text{y}}=\frac{1}{2}\big[\log\text{y}\big]^{10}_{5}=\frac{1}{2}\big[\log10-\log5\big]=\frac{1}{2}\log\bigg[\frac{10}{5}\bigg]=\frac{1}{2}\log2$

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