Evaluate the following integrals: ^ 2x dx - Vidyadip

Question

Evaluate the following integrals:

$\int\text{xe}^{2\text{x}}\text{dx}$

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Answer

Let $\text{I}=\int\text{xe}^{2\text{x}}\text{dx}$
Using integration by parts,
$\text{I}=\text{x}\int\text{e}^{2\text{x}}\text{dx}-\int(1\times\int\text{e}^{2\text{x}}\text{dx})\text{dx+C}$
$=\frac{\text{xe}^{2\text{x}}}{2}-\int\Big(\frac{\text{e}^{2\text{x}}}{2}\Big)\text{dx+C}$
$=\frac{\text{xe}^{2\text{x}}}{2}-\frac{\text{e}^{2\text{x}}}{4}+\text{C}$
$\text{I}=\Big(\frac{\text{x}}{2}-\frac{1}{4}\Big)\text{e}^{2\text{x}}+\text{C}$

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