What is the integral from 0 to infinity of xe^{-ax} ?

The integral from 0 to infinity of xe^{-ax} is 1/(a^2)

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integral from 0 to infinity of xe^{-ax}

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

\frac{1}{a ^{2}}

Solution steps

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

Compute the indefinite integral:

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

Compute the boundaries:

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

= 0 - (- \frac{1}{a ^{2}})

Simplify

= \frac{1}{a ^{2}}

\frac{d}{d x} (\frac{e ^{2 x} - 1}{1 - e ^{- 2 x}})

\frac{d}{d x} ((x^{2} + 17) (x^{3} - 3 x + 1))

y^{^{''}} + 9 y = 9 x^{2} + 9 x + 11

x \frac{d y}{d x} + y ln (x) - y = y ln (y)

\frac{\partial}{\partial y} (\frac{1}{x ^{2} yz})

What is the integral from 0 to infinity of xe^{-ax} ?
The integral from 0 to infinity of xe^{-ax} is 1/(a^2)