What is the integral of x^6e^x ?

The integral of x^6e^x is x^6e^x-6(x^5e^x-5(x^4e^x-4(x^3e^x-3(x^2e^x-2(e^xx-e^x)))))+C

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integral of x^6e^x

\int x^{6} e^{x} d x

x^{6} e^{x} - 6 (x^{5} e^{x} - 5 (x^{4} e^{x} - 4 (x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (e^{x} x - e^{x}))))) + C

Solution steps

\int x^{6} e^{x} d x

Apply Integration By Parts

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

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

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

Add a constant to the solution

= x^{6} e^{x} - 6 (x^{5} e^{x} - 5 (x^{4} e^{x} - 4 (x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (e^{x} x - e^{x}))))) + C

What is the integral of x^6e^x ?
The integral of x^6e^x is x^6e^x-6(x^5e^x-5(x^4e^x-4(x^3e^x-3(x^2e^x-2(e^xx-e^x)))))+C