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

The integral from 0 to infinity of axe^{-ax} is 1/a

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

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

\frac{1}{a}

Solution steps

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

Compute the indefinite integral:

\int a x e^{- a x} d x = - e^{- a x} x - \frac{e ^{- a x}}{a} + C

Compute the boundaries:

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

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

Simplify

= \frac{1}{a}

\int 2 x (x^{2} + 10)^{3} d x

\frac{d}{d x} (A e^{2 x})

\int e^{8 x} cos (5 x) d x

f (x) = 2 e^{x} + x, at x = 0

f (x) = t sin (t)

What is the integral from 0 to infinity of axe^{-ax} ?
The integral from 0 to infinity of axe^{-ax} is 1/a