The correct option is A e^x(x^2 2x+2)+c Let I=∫ x^2e^x #160;dx #160; = #160; x^2∫ #160;e^xdx ∫ (dx^2/dx·∫ e^x dx).dx =x^2 ...

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Combination

Evaluate ∫ ...

Question

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

e^{x} (x^{2} - 2 x + 2) + c

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e^{x} (x^{2} + 2 x + 2) + c

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x^{2} + e x + c

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e^{x} (x^{2} + x + 2) + c

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Solution

The correct option is A $e^{x} (x^{2} - 2 x + 2) + c$
Let $I = \int x^{2} e^{x} d x$
$= x^{2} \int e^{x} d x$ $- \int (\frac{d x^{2}}{d x} \cdot \int e^{x} d x) . d x$
$= x^{2} e^{x} - \int (2 x e^{x}) d x \cdot$
$= x^{2} e^{x} - 2 \int x e^{x} \cdot d x \cdot$
Taking $\int x e^{x} = x e^{x} - e^{x} \cdot$ ....(by using by parts)
Therefore, $I = x^{2} e^{x} - 2 [x e^{x} - e^{x}] + c$
$= e^{x} [x^{2} - 2 x + 2] + c$

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