Evaluate the definite integral in Exercise: _ 2 ^ 3 x dx x^ 2 +1 - Vidyadip

Question

Evaluate the definite integral in Exercise:
$\int\limits_{2}^{3}\frac{\text{x}\ \text{dx}}{\text{x}^{2}+1}$

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Answer

$\text{Let}\text{I}=\int\limits_{2}^{3}\frac{\text{x}}{\text{x}^{2}+1}\text{dx}$
$\int\frac{\text{x}}{\text{x}^{2}+1}\text{dx}=\frac{1}{2}\int\frac{2\text{x}}{\text{x}^{2}+1}\text{dx}=\frac{1}{2}\text{log}(1+\text{x}^{2})=\text{F}\text{(x)}$
By second fundamental theorem of calculus, we obtain
$\text{I}=\text{F}(3)-\text{F}(2)$
$=\frac{1}{2}\big[\text{log}\big(1+(3)^{2}\big)-\text{log}\big(1+(2)^{2}\big)\big]$
$=\frac{1}{2}\big[\text{log}(10)-\text{log(5)}\big]$
$=\frac{1}{2}\text{log}\bigg(\frac{10}{5}\bigg)=\frac{1}{2}\text{log}2$

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