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

The integral from 0 to infinity of x^2e^{-2x} is 1/4

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integral from 0 to infinity of x^2e^{-2x}

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

\frac{1}{4}

Decimal

0.25

Solution steps

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

Compute the indefinite integral:

\int x^{2} e^{- 2 x} d x = - \frac{1}{2} e^{- 2 x} x^{2} + \frac{1}{4} (- 2 e^{- 2 x} x - e^{- 2 x}) + C

Compute the boundaries:

\int_{0}^{\infty} x^{2} e^{- 2 x} d x = 0 - (- \frac{1}{4})

= 0 - (- \frac{1}{4})

Simplify

= \frac{1}{4}

What is the integral from 0 to infinity of x^2e^{-2x} ?
The integral from 0 to infinity of x^2e^{-2x} is 1/4