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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

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

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

Use the common integral:

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

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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