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

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

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

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

\frac{2}{3} e^{\frac{x ^{3}}{2}} + C

Solution steps

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

Apply u-substitution

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

= \int \frac{e ^{\frac{u}{2}}}{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^{\frac{u}{2}} d u

Apply u-substitution

: \int e^{v} \cdot 2 d v

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

Use the common integral:

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

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

Substitute back

= \frac{1}{3} \cdot 2 e^{\frac{x ^{3}}{2}}

Simplify

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

= \frac{2}{3} e^{\frac{x ^{3}}{2}}

Add a constant to the solution

= \frac{2}{3} e^{\frac{x ^{3}}{2}} + C

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