Evaluate the following definite integrals: _ 1 ^ 2 ( x-1 x^2 )e^ … - Vidyadip

Question

Evaluate the following definite integrals:
$\int\limits_{1}^{2}\Big(\frac{\text{x}-1}{\text{x}^2}\Big)\text{e}^{\text{x}}\text{ dx}$

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Answer

Let $\text{I}=\int_{1}^\limits{2}\Big(\frac{\text{x}-1}{\text{x}^2}\Big)\text{e}^{\text{x}}\text{ dx}$ Then,
$\text{I}=\int_{1}^\limits{2}\Big(\frac{\text{e}^{\text{x}}}{\text{x}}-\frac{\text{e}^{\text{x}}}{\text{x}^2}\Big)\text{dx}$
$\Rightarrow\text{I}=\int_{1}^\limits{2}\frac{\text{e}^\text{x}}{\text{x}}\text{ dx}-\int_{1}^\limits{2}\frac{\text{e}^\text{x}}{\text{x}^2}\text{ dx}$
Integrating first term by parts,
$\text{I}=\bigg\{\Big[\frac{\text{e}^\text{x}}{\text{x}}\Big]^2_1-\int_{1}^\limits{2}\frac{-1}{\text{x}^2}\text{e}^{\text{x}}\text{ dx}\bigg\}-\int_{1}^\limits{2}\frac{\text{e}^\text{x}}{\text{x}^2}\text{ dx}$
$\Rightarrow\text{I}=\Big[\frac{\text{e}^\text{x}}{\text{x}}\Big]^2_1+\int_{1}^\limits{2}\frac{-1}{\text{x}^2}\text{e}^{\text{x}}\text{ dx}-\int_{1}^\limits{2}\frac{\text{e}^\text{x}}{\text{x}^2}\text{ dx}$
$\Rightarrow\text{I}=\Big[\frac{\text{e}^\text{x}}{\text{x}}\Big]^2_1$
$\Rightarrow\text{I}=\frac{\text{e}^2}{2}-\text{e}$

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