What is the integral of 3x^3e^{3x^2} ?

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

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integral of 3x^3e^{3x^2}

\int 3 x^{3} e^{3 x^{2}} d x

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

Solution steps

\int 3 x^{3} e^{3 x^{2}} d x

Take the constant out:

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

= 3 \cdot \int x^{3} e^{3 x^{2}} d x

Apply u-substitution

: \int \frac{e ^{u} u}{18} d u

= 3 \cdot \int \frac{e ^{u} u}{18} d u

Take the constant out:

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

= 3 \cdot \frac{1}{18} \cdot \int e^{u} u d u

Apply Integration By Parts

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

= 3 \cdot \frac{1}{18} (e^{u} u - \int e^{u} d u)

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

= 3 \cdot \frac{1}{18} (e^{u} u - e^{u})

Substitute back

u = 3 x^{2}

= 3 \cdot \frac{1}{18} (e^{3 x^{2}} \cdot 3 x^{2} - e^{3 x^{2}})

Simplify

3 \cdot \frac{1}{18} (e^{3 x^{2}} \cdot 3 x^{2} - e^{3 x^{2}}) : \frac{1}{6} (3 e^{3 x^{2}} x^{2} - e^{3 x^{2}})

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

Add a constant to the solution

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

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