What is the integral of 6x^3e^{-5x} ?

The integral of 6x^3e^{-5x} is 6(-1/5 e^{-5x}x^3-3/625 (25e^{-5x}x^2-2(-5e^{-5x}x-e^{-5x})))+C

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integral of 6x^3e^{-5x}

\int 6 x^{3} e^{- 5 x} d x

6 (- \frac{1}{5} e^{- 5 x} x^{3} - \frac{3}{625} (25 e^{- 5 x} x^{2} - 2 (- 5 e^{- 5 x} x - e^{- 5 x}))) + C

Solution steps

\int 6 x^{3} e^{- 5 x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 6 \cdot \int x^{3} e^{- 5 x} d x

Apply Integration By Parts

: - \frac{1}{5} e^{- 5 x} x^{3} - \int - \frac{3}{5} e^{- 5 x} x^{2} d x

= 6 (- \frac{1}{5} e^{- 5 x} x^{3} - \int - \frac{3}{5} e^{- 5 x} x^{2} d x)

\int - \frac{3}{5} e^{- 5 x} x^{2} d x = \frac{3}{625} (25 e^{- 5 x} x^{2} - 2 (- 5 e^{- 5 x} x - e^{- 5 x}))

= 6 (- \frac{1}{5} e^{- 5 x} x^{3} - \frac{3}{625} (25 e^{- 5 x} x^{2} - 2 (- 5 e^{- 5 x} x - e^{- 5 x})))

Add a constant to the solution

= 6 (- \frac{1}{5} e^{- 5 x} x^{3} - \frac{3}{625} (25 e^{- 5 x} x^{2} - 2 (- 5 e^{- 5 x} x - e^{- 5 x}))) + C

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What is the integral of 6x^3e^{-5x} ?
The integral of 6x^3e^{-5x} is 6(-1/5 e^{-5x}x^3-3/625 (25e^{-5x}x^2-2(-5e^{-5x}x-e^{-5x})))+C