Evaluate the following integrals: ^xdx - Vidyadip

Question

Evaluate the following integrals: $\int\text{xe}^\text{x}\text{dx}$

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Answer

$\int\text{xe}^{\text{x}}\text{dx}$
Taking x as the first function and $e^x$ as the second function.
$=\text{x}\int\text{e}^{\text{x}}\text{dx}-\int\Big\{\frac{\text{d}}{\text{dx}}(\text{x})\int\text{e}^{\text{x}}\text{dx}\Big\}\text{dx}$
$=\text{x}\text{e}^{\text{x}}-\int1(\text{e}^{\text{x}})\text{dx}$
$=\text{xe}^{\text{x}}-\text{e}^{\text{x}}+\text{C}$
$=(\text{x}-1)\text{e}^{\text{x}}+\text{C}$

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