What is the integral of 6x^3e^{x^2+3} ?

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

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integral of 6x^3e^{x^2+3}

\int 6 x^{3} e^{x^{2} + 3} d x

3 e^{3} (e^{x^{2}} x^{2} - e^{x^{2}}) + C

Solution steps

\int 6 x^{3} e^{x^{2} + 3} d x

Take the constant out:

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

= 6 \cdot \int x^{3} e^{x^{2} + 3} d x

x^{3} e^{x^{2} + 3} = e^{3} x^{3} e^{x^{2}}

= 6 \cdot \int e^{3} x^{3} e^{x^{2}} d x

Take the constant out:

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

= 6 e^{3} \cdot \int x^{3} e^{x^{2}} d x

Apply u-substitution

= 6 e^{3} \cdot \int \frac{e ^{u^{\frac{2}{3}}} u ^{\frac{1}{3}}}{3} d u

Take the constant out:

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

= 6 e^{3} \frac{1}{3} \cdot \int e^{u^{\frac{2}{3}}} u^{\frac{1}{3}} d u

Apply u-substitution

= 6 e^{3} \frac{1}{3} \cdot \int \frac{3 e ^{v} v}{2} d v

Take the constant out:

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

= 6 e^{3} \frac{1}{3} \cdot \frac{3}{2} \cdot \int e^{v} v d v

Apply Integration By Parts

= 6 e^{3} \frac{1}{3} \cdot \frac{3}{2} (e^{v} v - \int e^{v} d v)

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

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

Substitute back

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

Simplify

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

= 3 e^{3} (e^{x^{2}} x^{2} - e^{x^{2}})

Add a constant to the solution

= 3 e^{3} (e^{x^{2}} x^{2} - e^{x^{2}}) + C

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