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

Question

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

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Answer

Let $\text{I}=\int_{1}^\limits{2}\log\text{x}\text{ dx}$ Then,
$\text{I}=\int_{1}^\limits{2}1\log\text{x}\text{ dx}$
Integrating by parts.
$\Rightarrow\text{I}=\big[\text{x }\log\text{ x}\big]^2_1-\int_{1}^\limits{2}\frac{1}{\text{x}}\text{x}\text{ dx}$
$\Rightarrow\text{I}=\big[\text{x }\log\text{ x}\big]^2_1-\int_{1}^\limits{2}\text{dx}$
$\Rightarrow\text{I}=\big[\text{x }\log\text{ x}\big]^2_1-\big[\text{x}\big]^2_1$
$\Rightarrow\text{I}=2\log2-2+1$
$\Rightarrow\text{I}=2\log2-1$

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