Mathematics

$$\int { x{ e }^{ 2x }dx } = $$


SOLUTION

Consider the given integral.

$$I=\int{x{{e}^{2x}}}dx$$

We know that

$$\int{uvdx=u\int{vdx-\int{\left( \dfrac{d\left( u \right)}{dx}\int{vdx} \right)}}}dx$$

Therefore,

$$ I=x\left( \dfrac{{{e}^{2x}}}{2} \right)-\int{1\left( \dfrac{{{e}^{2x}}}{2} \right)}dx $$

$$ I=\dfrac{x{{e}^{2x}}}{2}-\dfrac{1}{2}\int{{{e}^{2x}}}dx $$

$$ I=\dfrac{x{{e}^{2x}}}{2}-\dfrac{1}{2}\left( \dfrac{{{e}^{2x}}}{2} \right)+C $$

$$ I=\dfrac{x{{e}^{2x}}}{2}-\dfrac{{{e}^{2x}}}{4}+C $$

$$ I=\dfrac{{{e}^{2x}}\left( 2x-1 \right)}{4}+C $$

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Subjective Medium Published on 17th 09, 2020
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