Prove the following Exercise: ^ 1 _ 0 x e^ x dx=1 - Vidyadip

Question

Prove the following Exercise:
$\int^{1}_{0}\text{x e}^{\text{x}}\ \text{dx}=1$

✓

Answer

$\text{Let I}=\int^{1}_{0}\text{x e}^{\text{x}}\ \text{dx}$
Integrating by parts, we obtain
$\text{I}=\text{x}\int^{1}_{0}\text{e}^{\text{x}}\ \text{dx}-\left\{\bigg(\frac{\text{d}}{\text{dx}}\text{(x)}\bigg)\int\text{e}^{\text{x}}\text{dx}\right\}\text{dx}$
$=\Big[\text{xe}^{\text{x}}\Big]^{1}_{0}-\int^{1}_{0}\text{e}^{\text{x}}\ \text{dx}$
$=\Big[\text{xe}^{\text{x}}\Big]^{1}_{0}-\Big[\text{e}^{\text{x}}\Big]^{1}_{0}$
$=\text{e}-\text{e}+1$
$=1$
Hence, the given result is Proved.

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