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Calculus Level 2

∫ 0 ∞ x 2 e x d x = ? \displaystyle\int_{0}^{\infty}{\dfrac{{x}^{2}}{{e}^{x}}dx} \ = \ ?


The answer is 2.

Hamza A by Hamza A , 19, USA

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2 solutions

∫ 0 ∞ x 2 e x d x = Γ [ 3 ] = 2 ! = 2 \int_{0}^\infty \frac {x^2}{e^x} dx = \Gamma[3] = 2! =2 You can do this integral by parts twice, u = x 2 , v = − e − x , d v = e − x d x . . . u = x^2, \space v = -e^{-x},\space dv = e^{-x} dx...

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∫ 0 ∞ x 2 e x d x = 2 ( − e − x x − e − x ) − e x x 2 + C \int_{0}^{\infty}\frac{x^2}{e^x}\,dx = 2(-e^{-x}x-e^{-x})-e^xx^2+C

Use: ∫ a b f ( x ) d x \int_{a}^{b}f(x)\,dx

ADIOS!!! \LARGE \text{ADIOS!!!}

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