What is the integral from 0 to infinity of x^3e^{-3x} ?

The integral from 0 to infinity of x^3e^{-3x} is 2/27

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integral from 0 to infinity of x^3e^{-3x}

\int_{0}^{\infty} x^{3} e^{- 3 x} d x

\frac{2}{27}

Decimal

0.07407 \dots

Solution steps

\int_{0}^{\infty} x^{3} e^{- 3 x} d x

Compute the indefinite integral:

\int x^{3} e^{- 3 x} d x = - \frac{1}{3} e^{- 3 x} x^{3} - \frac{1}{27} (9 e^{- 3 x} x^{2} - 2 (- 3 e^{- 3 x} x - e^{- 3 x})) + C

Compute the boundaries:

\int_{0}^{\infty} x^{3} e^{- 3 x} d x = 0 - (- \frac{2}{27})

= 0 - (- \frac{2}{27})

Simplify

= \frac{2}{27}

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What is the integral from 0 to infinity of x^3e^{-3x} ?
The integral from 0 to infinity of x^3e^{-3x} is 2/27