What is the integral from 0 to infinity of x^2e^{(-x)/θ} ?

The integral from 0 to infinity of x^2e^{(-x)/θ} is 2θ^3

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integral from 0 to infinity of x^2e^{(-x)/θ}

\int_{0}^{\infty} x^{2} e^{\frac{- x}{θ}} d x

2 θ^{3}

Solution steps

\int_{0}^{\infty} x^{2} e^{\frac{- x}{θ}} d x

Compute the indefinite integral:

\int x^{2} e^{\frac{- x}{θ}} d x = - θ e^{- \frac{x}{θ}} x^{2} + 2 θ (- θ e^{- \frac{x}{θ}} x - θ^{2} e^{- \frac{x}{θ}}) + C

Compute the boundaries:

\int_{0}^{\infty} x^{2} e^{\frac{- x}{θ}} d x = 0 - (- 2 θ^{3})

= 0 - (- 2 θ^{3})

Simplify

= 2 θ^{3}

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What is the integral from 0 to infinity of x^2e^{(-x)/θ} ?
The integral from 0 to infinity of x^2e^{(-x)/θ} is 2θ^3