What is the integral from 0 to infinty of y^5e^{-5y} ?

The integral from 0 to infinty of y^5e^{-5y} is

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integral from 0 to infinty of y^5e^{-5y}

\int_{0}^{in f in t y} y^{5} e^{- 5 y} d y

- \frac{1}{3125} e^{- 5 n^{2} f i^{2} y t} (625 n^{10} f^{5} i^{10} y^{5} t^{5} + 625 n^{8} f^{4} i^{8} y^{4} t^{4} + 4 (125 n^{6} f^{3} i^{6} y^{3} t^{3} + 3 (25 n^{4} f^{2} i^{4} y^{2} t^{2} + 2 (5 n^{2} f i^{2} y t + 1)))) + \frac{24}{3125}

Solution steps

\int_{0}^{in f in t y} y^{5} e^{- 5 y} d y

Compute the indefinite integral:

\int y^{5} e^{- 5 y} d y = - \frac{1}{3125} e^{- 5 y} (625 y^{5} + 625 y^{4} + 4 (125 y^{3} + 3 (25 y^{2} + 2 (5 y + 1)))) + C

Compute the boundaries:

\int_{0}^{in f in t y} y^{5} e^{- 5 y} d y = - \frac{1}{3125} e^{- 5 n^{2} f i^{2} y t} (625 n^{10} f^{5} i^{10} y^{5} t^{5} + 625 n^{8} f^{4} i^{8} y^{4} t^{4} + 4 (125 n^{6} f^{3} i^{6} y^{3} t^{3} + 3 (25 n^{4} f^{2} i^{4} y^{2} t^{2} + 2 (5 n^{2} f i^{2} y t + 1)))) - (- \frac{24}{3125})

= - \frac{1}{3125} e^{- 5 n^{2} f i^{2} y t} (625 n^{10} f^{5} i^{10} y^{5} t^{5} + 625 n^{8} f^{4} i^{8} y^{4} t^{4} + 4 (125 n^{6} f^{3} i^{6} y^{3} t^{3} + 3 (25 n^{4} f^{2} i^{4} y^{2} t^{2} + 2 (5 n^{2} f i^{2} y t + 1)))) - (- \frac{24}{3125})

Simplify

= - \frac{1}{3125} e^{- 5 n^{2} f i^{2} y t} (625 n^{10} f^{5} i^{10} y^{5} t^{5} + 625 n^{8} f^{4} i^{8} y^{4} t^{4} + 4 (125 n^{6} f^{3} i^{6} y^{3} t^{3} + 3 (25 n^{4} f^{2} i^{4} y^{2} t^{2} + 2 (5 n^{2} f i^{2} y t + 1)))) + \frac{24}{3125}

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What is the integral from 0 to infinty of y^5e^{-5y} ?
The integral from 0 to infinty of y^5e^{-5y} is