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

The integral from 0 to infinity of x*e^{-3x} is 1/9

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

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

\frac{1}{9}

Decimal

0.11111 \dots

Solution steps

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

Compute the indefinite integral:

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

Compute the boundaries:

\int_{0}^{\infty} x e^{- 3 x} d x = 0 - (- \frac{1}{9})

= 0 - (- \frac{1}{9})

Simplify

= \frac{1}{9}

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What is the integral from 0 to infinity of x*e^{-3x} ?
The integral from 0 to infinity of x*e^{-3x} is 1/9