The correct option is B (xe^x)^2/2 I=∫ xe^2x(1+x)dx =∫ xe^2xdx+∫ x^2e^2xdx =e^2xx^2/2 ∫ #160;x^2e^2xdx+∫ #160;x^2e^2xdx I=(xe^x)^2 ...

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Substitution Method to Remove Indeterminate Form

∫ xe2x1+xdx e...

Question

$\int x e^{2 x} (1 + x) d x$ equal to

\frac{x e^{x}}{2} + c

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\frac{(e^{x})^{2}}{2} r

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\frac{(1 + x)^{2}}{2} + c

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\frac{(x e^{x})^{2}}{2}

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Solution

The correct option is B $\frac{(x e^{x})^{2}}{2}$
$I = \int x e^{2 x} (1 + x) d x$
$= \int x e^{2 x} d x + \int x^{2} e^{2 x} d x$
$= e^{2 x} \frac{x^{2}}{2} - \int x^{2} e^{2 x} d x + \int x^{2} e^{2 x} d x$
$I = \frac{(x e^{x})^{2}}{2}$

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