Evaluate the following definite integrals: _ 1 ^ e e^ x x (1+x )d… - Vidyadip

Question

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

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Answer

Let $\text{I}=\int_{1}^\limits{\text{e}}\frac{\text{e}^{\text{x}}}{\text{x}}(1+\text{x}\log\text{x})\text{dx}$$\text{I}=\int_{1}^\limits{\text{e}}\frac{\text{e}^{\text{x}}}{\text{x}}+\int_{1}^\limits{\text{e}}\text{e}^{\text{x}}\log\text{x}\text{ dx}$
$\text{I}=\big[\text{e}^{\text{x}}\log\text{x}\big]^{\text{e}}_1-\int_{1}^\limits{\text{e}}\text{e}^{\text{x}}\log\text{x}+\int_{1}^\limits{\text{e}}\text{e}^{\text{x}}\log\text{x}$
$\text{I}=\big[\text{e}^{\text{x}}\log\text{x}\big]^{\text{e}}_1$
$\text{I}=\big[\text{e}^{\text{x}}\log\text{e}-\text{e}^1\log1\big]$
$\text{I}=\big[\text{e}^{\text{e}}1-0\big]$
$\text{I}=\text{e}^{\text{e}}$

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