What is the integral from 0 to 1 of x^6*e^{-x} ?

The integral from 0 to 1 of x^6*e^{-x} is -(1957)/e+720

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integral from 0 to 1 of x^6*e^{-x}

\int_{0}^{1} x^{6} \cdot e^{- x} d x

- \frac{1957}{e} + 720

Decimal

0.05993 \dots

Solution steps

\int_{0}^{1} x^{6} e^{- x} d x

Apply Integration By Parts

: [- e^{- x} x^{6} - \int - 6 e^{- x} x^{5} d x]_{0}^{1}

= [- e^{- x} x^{6} - \int - 6 e^{- x} x^{5} d x]_{0}^{1}

\int - 6 e^{- x} x^{5} d x = - 6 (- e^{- x} x^{5} - 5 (e^{- x} x^{4} - 4 (- e^{- x} x^{3} - 3 (e^{- x} x^{2} - 2 (- e^{- x} x - e^{- x})))))

= [- e^{- x} x^{6} - (- 6 (- e^{- x} x^{5} - 5 (e^{- x} x^{4} - 4 (- e^{- x} x^{3} - 3 (e^{- x} x^{2} - 2 (- e^{- x} x - e^{- x}))))))]_{0}^{1}

Simplify

= [- e^{- x} x^{6} + 6 (- e^{- x} x^{5} - 5 (e^{- x} x^{4} - 4 (- e^{- x} x^{3} - 3 (e^{- x} x^{2} - 2 (- e^{- x} x - e^{- x})))))]_{0}^{1}

Compute the boundaries:

- \frac{1957}{e} + 720

= - \frac{1957}{e} + 720

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What is the integral from 0 to 1 of x^6*e^{-x} ?
The integral from 0 to 1 of x^6*e^{-x} is -(1957)/e+720