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

The integral from 0 to infinity of x^3e^{-x} is 6

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

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

6

Solution steps

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

Compute the indefinite integral:

\int x^{3} e^{- x} d x = - e^{- x} x^{3} - 3 (e^{- x} x^{2} - 2 (- e^{- x} x - e^{- x})) + C

Compute the boundaries:

\int_{0}^{\infty} x^{3} e^{- x} d x = 0 - (- 6)

= 0 - (- 6)

Simplify

= 6

\frac{d}{d x} (sech (x))

x^{3} \cdot e^{x}

x \to 1 lim (- \frac{2}{x ^{3}})

\int \frac{7 x ^{6}}{( 4 + x ^{7} ) ^{5}} d x

\int sin (5) (t) cos (6) (t) d t

What is the integral from 0 to infinity of x^3e^{-x} ?
The integral from 0 to infinity of x^3e^{-x} is 6