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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{e ^{u^{\frac{2}{3}}} u ^{\frac{1}{3}}}{3} d u

= \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

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

Apply u-substitution

: \int \frac{3 e ^{v} v}{2} d v

= \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

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

Apply Integration By Parts

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

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

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

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

Substitute back

= \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

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

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

Add a constant to the solution

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

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