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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

Apply u-substitution

: \int e^{v} \cdot 3 d v

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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