What is the integral of x^4e^{5x} ?

The integral of x^4e^{5x} is 1/5 e^{5x}x^4-4/3125 (125e^{5x}x^3-3(25e^{5x}x^2-2(5e^{5x}x-e^{5x})))+C

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integral of x^4e^{5x}

\int x^{4} e^{5 x} d x

\frac{1}{5} e^{5 x} x^{4} - \frac{4}{3125} (125 e^{5 x} x^{3} - 3 (25 e^{5 x} x^{2} - 2 (5 e^{5 x} x - e^{5 x}))) + C

Solution steps

\int x^{4} e^{5 x} d x

Apply Integration By Parts

= \frac{1}{5} e^{5 x} x^{4} - \int \frac{4}{5} e^{5 x} x^{3} d x

\int \frac{4}{5} e^{5 x} x^{3} d x = \frac{4}{3125} (125 e^{5 x} x^{3} - 3 (25 e^{5 x} x^{2} - 2 (5 e^{5 x} x - e^{5 x})))

= \frac{1}{5} e^{5 x} x^{4} - \frac{4}{3125} (125 e^{5 x} x^{3} - 3 (25 e^{5 x} x^{2} - 2 (5 e^{5 x} x - e^{5 x})))

Add a constant to the solution

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

What is the integral of x^4e^{5x} ?
The integral of x^4e^{5x} is 1/5 e^{5x}x^4-4/3125 (125e^{5x}x^3-3(25e^{5x}x^2-2(5e^{5x}x-e^{5x})))+C