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

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

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

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

9

Solution steps

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

Compute the indefinite integral:

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

Compute the boundaries:

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

= 0 - (- 9)

Simplify

= 9

x^{2}, 6 + x

\int sin^{3} (\frac{x}{10}) cos^{2} (\frac{x}{10}) d x

y^{^{''}} + 12 y^{^{'}} + 35 y = 70 t^{2} + 48 t + 4 + 48 e^{t}

h (t) = - 5 t^{2}

\frac{d}{d x} (\frac{5 sin ( x )}{3 + cos ( x )})

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