What is the integral of x^8xe^{x^2} ?

The integral of x^8xe^{x^2} is 1/2 e^{x^2}x^8-2e^{x^2}x^6+6e^{x^2}x^4-12e^{x^2}x^2+12e^{x^2}+C

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integral of x^8xe^{x^2}

\int x^{8} x e^{x^{2}} d x

\frac{1}{2} e^{x^{2}} x^{8} - 2 e^{x^{2}} x^{6} + 6 e^{x^{2}} x^{4} - 12 e^{x^{2}} x^{2} + 12 e^{x^{2}} + C

Solution steps

\int x^{8} x e^{x^{2}} d x

Simplify

x^{8} x : x^{9}

= \int x^{9} e^{x^{2}} d x

Apply Integration By Parts

: \frac{1}{10} e^{x^{2}} x^{10} - \int \frac{1}{5} e^{x^{2}} x^{11} d x

= \frac{1}{10} e^{x^{2}} x^{10} - \int \frac{1}{5} e^{x^{2}} x^{11} d x

\int \frac{1}{5} e^{x^{2}} x^{11} d x = \frac{1}{10} (e^{x^{2}} x^{10} - 5 (e^{x^{2}} x^{8} - 4 (e^{x^{2}} x^{6} - 3 (e^{x^{2}} x^{4} - 2 (e^{x^{2}} x^{2} - e^{x^{2}})))))

= \frac{1}{10} e^{x^{2}} x^{10} - \frac{1}{10} (e^{x^{2}} x^{10} - 5 (e^{x^{2}} x^{8} - 4 (e^{x^{2}} x^{6} - 3 (e^{x^{2}} x^{4} - 2 (e^{x^{2}} x^{2} - e^{x^{2}})))))

Simplify

\frac{1}{10} e^{x^{2}} x^{10} - \frac{1}{10} (e^{x^{2}} x^{10} - 5 (e^{x^{2}} x^{8} - 4 (e^{x^{2}} x^{6} - 3 (e^{x^{2}} x^{4} - 2 (e^{x^{2}} x^{2} - e^{x^{2}}))))) : \frac{1}{2} e^{x^{2}} x^{8} - 2 e^{x^{2}} x^{6} + 6 e^{x^{2}} x^{4} - 12 e^{x^{2}} x^{2} + 12 e^{x^{2}}

= \frac{1}{2} e^{x^{2}} x^{8} - 2 e^{x^{2}} x^{6} + 6 e^{x^{2}} x^{4} - 12 e^{x^{2}} x^{2} + 12 e^{x^{2}}

Add a constant to the solution

= \frac{1}{2} e^{x^{2}} x^{8} - 2 e^{x^{2}} x^{6} + 6 e^{x^{2}} x^{4} - 12 e^{x^{2}} x^{2} + 12 e^{x^{2}} + C

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What is the integral of x^8xe^{x^2} ?
The integral of x^8xe^{x^2} is 1/2 e^{x^2}x^8-2e^{x^2}x^6+6e^{x^2}x^4-12e^{x^2}x^2+12e^{x^2}+C