What is the integral of x^5e^{3x^3} ?

The integral of x^5e^{3x^3} is 1/27 (3e^{3x^3}x^3-e^{3x^3})+C

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integral of x^5e^{3x^3}

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

\frac{1}{27} (3 e^{3 x^{3}} x^{3} - e^{3 x^{3}}) + C

Solution steps

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

Apply u-substitution

: \int \frac{e ^{3 u^{\frac{3}{5}}} u ^{\frac{1}{5}}}{5} d u

= \int \frac{e ^{3 u^{\frac{3}{5}}} u ^{\frac{1}{5}}}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int e^{3 u^{\frac{3}{5}}} u^{\frac{1}{5}} d u

Apply u-substitution

: \int \frac{5 e ^{3 v} v}{3} d v

= \frac{1}{5} \cdot \int \frac{5 e ^{3 v} v}{3} d v

Take the constant out:

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

= \frac{1}{5} \cdot \frac{5}{3} \cdot \int e^{3 v} v d v

Apply u-substitution

: \int \frac{e ^{w} w}{9} d w

= \frac{1}{5} \cdot \frac{5}{3} \cdot \int \frac{e ^{w} w}{9} d w

Take the constant out:

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

= \frac{1}{5} \cdot \frac{5}{3} \cdot \frac{1}{9} \cdot \int e^{w} w d w

Apply Integration By Parts

: e^{w} w - \int e^{w} d w

= \frac{1}{5} \cdot \frac{5}{3} \cdot \frac{1}{9} (e^{w} w - \int e^{w} d w)

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

= \frac{1}{5} \cdot \frac{5}{3} \cdot \frac{1}{9} (e^{w} w - e^{w})

Substitute back

= \frac{1}{5} \cdot \frac{5}{3} \cdot \frac{1}{9} (e^{3 (x^{5})^{\frac{3}{5}}} \cdot 3 (x^{5})^{\frac{3}{5}} - e^{3 (x^{5})^{\frac{3}{5}}})

Simplify

\frac{1}{5} \cdot \frac{5}{3} \cdot \frac{1}{9} (e^{3 (x^{5})^{\frac{3}{5}}} \cdot 3 (x^{5})^{\frac{3}{5}} - e^{3 (x^{5})^{\frac{3}{5}}}) : \frac{1}{27} (3 e^{3 x^{3}} x^{3} - e^{3 x^{3}})

= \frac{1}{27} (3 e^{3 x^{3}} x^{3} - e^{3 x^{3}})

Add a constant to the solution

= \frac{1}{27} (3 e^{3 x^{3}} x^{3} - e^{3 x^{3}}) + C

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What is the integral of x^5e^{3x^3} ?
The integral of x^5e^{3x^3} is 1/27 (3e^{3x^3}x^3-e^{3x^3})+C