What is the integral from 0 to 1 of x^3e^{-4x} ?

The integral from 0 to 1 of x^3e^{-4x} is -(71)/(128e^4)+3/128

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integral from 0 to 1 of x^3e^{-4x}

\int_{0}^{1} x^{3} e^{- 4 x} d x

- \frac{71}{128 e ^{4}} + \frac{3}{128}

Decimal

0.01327 \dots

Solution steps

\int_{0}^{1} x^{3} e^{- 4 x} d x

Apply Integration By Parts

: [- \frac{1}{4} e^{- 4 x} x^{3} - \int - \frac{3}{4} e^{- 4 x} x^{2} d x]_{0}^{1}

= [- \frac{1}{4} e^{- 4 x} x^{3} - \int - \frac{3}{4} e^{- 4 x} x^{2} d x]_{0}^{1}

\int - \frac{3}{4} e^{- 4 x} x^{2} d x = \frac{3}{128} e^{- 4 x} (8 x^{2} + 4 x + 1)

= [- \frac{1}{4} e^{- 4 x} x^{3} - \frac{3}{128} e^{- 4 x} (8 x^{2} + 4 x + 1)]_{0}^{1}

Compute the boundaries:

- \frac{71}{128 e ^{4}} + \frac{3}{128}

= - \frac{71}{128 e ^{4}} + \frac{3}{128}

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What is the integral from 0 to 1 of x^3e^{-4x} ?
The integral from 0 to 1 of x^3e^{-4x} is -(71)/(128e^4)+3/128