What is the integral of 9x^5e^{8x^2} ?

The integral of 9x^5e^{8x^2} is 9/512 e^{8x^2}(32x^4-8x^2+1)+C

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integral of 9x^5e^{8x^2}

\int 9 x^{5} e^{8 x^{2}} d x

\frac{9}{512} e^{8 x^{2}} (32 x^{4} - 8 x^{2} + 1) + C

Solution steps

\int 9 x^{5} e^{8 x^{2}} d x

Take the constant out:

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

= 9 \cdot \int x^{5} e^{8 x^{2}} d x

Apply Integration By Parts

: \frac{1}{6} e^{8 x^{2}} x^{6} - \int \frac{8}{3} e^{8 x^{2}} x^{7} d x

= 9 (\frac{1}{6} e^{8 x^{2}} x^{6} - \int \frac{8}{3} e^{8 x^{2}} x^{7} d x)

\int \frac{8}{3} e^{8 x^{2}} x^{7} d x = \frac{1}{3072} (512 e^{8 x^{2}} x^{6} - 3 (64 e^{8 x^{2}} x^{4} - 2 (8 e^{8 x^{2}} x^{2} - e^{8 x^{2}})))

= 9 (\frac{1}{6} e^{8 x^{2}} x^{6} - \frac{1}{3072} (512 e^{8 x^{2}} x^{6} - 3 (64 e^{8 x^{2}} x^{4} - 2 (8 e^{8 x^{2}} x^{2} - e^{8 x^{2}}))))

Simplify

9 (\frac{1}{6} e^{8 x^{2}} x^{6} - \frac{1}{3072} (512 e^{8 x^{2}} x^{6} - 3 (64 e^{8 x^{2}} x^{4} - 2 (8 e^{8 x^{2}} x^{2} - e^{8 x^{2}})))) : \frac{9}{512} e^{8 x^{2}} (32 x^{4} - 8 x^{2} + 1)

= \frac{9}{512} e^{8 x^{2}} (32 x^{4} - 8 x^{2} + 1)

Add a constant to the solution

= \frac{9}{512} e^{8 x^{2}} (32 x^{4} - 8 x^{2} + 1) + C

\frac{\partial}{\partial y} (x^{3} y^{2})

\int cos (3 θ) d θ

x \to - 2 lim (\frac{3 x - 6}{x + 2})

\frac{d}{d x} (\frac{x ^{3}}{2 ( x + 1 ) ^{2}})

e^{x}

What is the integral of 9x^5e^{8x^2} ?
The integral of 9x^5e^{8x^2} is 9/512 e^{8x^2}(32x^4-8x^2+1)+C