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
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109

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