What is the integral of x^8e^{x^3} ?

The integral of x^8e^{x^3} is 1/3 (e^{x^3}x^6-2(e^{x^3}x^3-e^{x^3}))+C

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integral of x^8e^{x^3}

\int x^{8} e^{x^{3}} d x

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

Solution steps

\int x^{8} e^{x^{3}} d x

Apply u-substitution

: \int \frac{ln ^{2} ( u )}{3} d u

= \int \frac{ln ^{2} ( u )}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int ln^{2} (u) d u

Apply Integration By Parts

: u ln^{2} (u) - \int 2 ln (u) d u

= \frac{1}{3} (u ln^{2} (u) - \int 2 ln (u) d u)

\int 2 ln (u) d u = 2 (u ln (u) - u)

= \frac{1}{3} (u ln^{2} (u) - 2 (u ln (u) - u))

Substitute back

u = e^{x^{3}}

= \frac{1}{3} (e^{x^{3}} ln^{2} (e^{x^{3}}) - 2 (e^{x^{3}} ln (e^{x^{3}}) - e^{x^{3}}))

Simplify

\frac{1}{3} (e^{x^{3}} ln^{2} (e^{x^{3}}) - 2 (e^{x^{3}} ln (e^{x^{3}}) - e^{x^{3}})) : \frac{1}{3} (e^{x^{3}} x^{6} - 2 (e^{x^{3}} x^{3} - e^{x^{3}}))

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

Add a constant to the solution

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

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What is the integral of x^8e^{x^3} ?
The integral of x^8e^{x^3} is 1/3 (e^{x^3}x^6-2(e^{x^3}x^3-e^{x^3}))+C