I=∫x^2e^xdx Integrating by parts, we getLet u=x^2⇒ du=2x dx dv=e^xdx⇒ v=e^x I=x^2e^x ∫2xe^xdx I=x^2e^x 2∫xe^xdx Let u=x⇒ du=dx ...

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Chain Rule of Differentiation

∫x2 ex dx =?

Question

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

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Solution

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

Integrating by parts, we get

Let $u = x^{2} \Rightarrow d u = 2 x d x$

$d v = e^{x} d x \Rightarrow v = e^{x}$

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

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

Let $u = x \Rightarrow d u = d x$

$d v = e^{x} d x \Rightarrow v = e^{x}$

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

$I = x^{2} e^{x} - 2 [x e^{x} - e^{x}] + c$

$∴ I = x^{2} e^{x} - 2 x e^{x} + 2 e^{x} + c$

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