What is the integral from 0 to infinity of kte^{-kt} ?

The integral from 0 to infinity of kte^{-kt} is 1/k

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integral from 0 to infinity of kte^{-kt}

\int_{0}^{\infty} k t e^{- k t} d t

\frac{1}{k}

Solution steps

\int_{0}^{\infty} k t e^{- k t} d t

Compute the indefinite integral:

\int k t e^{- k t} d t = - e^{- k t} t - \frac{e ^{- k t}}{k} + C

Compute the boundaries:

\int_{0}^{\infty} k t e^{- k t} d t = 0 - (- \frac{1}{k})

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

Simplify

= \frac{1}{k}

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What is the integral from 0 to infinity of kte^{-kt} ?
The integral from 0 to infinity of kte^{-kt} is 1/k