Consider the given integral. #10; #10; I=∫xe^2xdx #10; #10;We know that #10; #10; ∫uvdx=u∫vdx ∫( d( u #10;)dx∫vdx)dx #10; #10;Theref ...

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Integration by Parts

∫ x e 2x dx =

Question

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

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Solution

Consider the given integral.

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

We know that

$\int u v d x = u \int v d x - \int (\frac{d (u)}{d x} \int v d x) d x$

Therefore,

$I = x (\frac{e^{2 x}}{2}) - \int 1 (\frac{e^{2 x}}{2}) d x$

$I = \frac{x e^{2 x}}{2} - \frac{1}{2} \int e^{2 x} d x$

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

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

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

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