Evaluate the following integrals: ^ 4 _2 x x^2+1 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^{4}_2\frac{\text{x}}{\text{x}^2+1}\text{dx}$

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Answer

$\int\limits^{4}_2\frac{\text{x}}{\text{x}^2+1}\text{dx}$
$=\frac{1}{2}\int\limits^{4}_2\frac{2\text{x}}{\text{x}^2+1}\text{dx}$
$=\frac{1}{2}\times\Big[\log(\text{x})^2-1\Big]^4_2$ $\Big[\int\frac{\text{f}'(\text{x})}{\text{f(x)}}\text{ dx}=\log\text{f(x)}+\text{C}\Big]$
$=\frac{1}{2}\big(\log17-\log5\big)$
$=\frac{1}{2}\log\Big(\frac{17}{5}\Big)$ $\Big(\log\text{a}-\log\text{b}=\log\frac{\text{a}}{\text{b}}\Big)$

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