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

The integral of x^5e^{xxx} is 1/3 (e^{x^3}x^3-e^{x^3})+C

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integral of x^5e^{xxx}

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

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

Solution steps

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

Simplify

xxx : x^{3}

= \int e^{x^{3}} x^{5} d x

Apply u-substitution

= \int \frac{e ^{u} 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 e^{u} u d u

Apply Integration By Parts

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

\int e^{u} d u = e^{u}

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

Substitute back

u = x^{3}

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

Add a constant to the solution

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

What is the integral of x^5e^{xxx} ?
The integral of x^5e^{xxx} is 1/3 (e^{x^3}x^3-e^{x^3})+C