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

Question

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

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Answer

Let $\text{I}=\int_{1}^\limits{\text{e}}\frac{\log\text{x}}{\text{x}}\text{ dx}$
Let $\log\text{x}=\text{u}$
$\Rightarrow\frac{1}{\text{x}}=\text{dx}=\text{du}$
$\therefore\text{ I}=\int\text{u}\text{ du}$
$\Rightarrow\text{I}=\Big[\frac{\text{u}^2}{2}\Big]$
$\Rightarrow\text{I}=\Big[\frac{(\log\text{x})}{2}\Big]^{\text{e}}_1$
$\Rightarrow\text{I}=\frac{1}{2}-0$
$\Rightarrow\text{I}=\frac{1}{2}$

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