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

The integral of xe^{3x^2+2} is 1/6 e^{3x^2+2}+C

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integral of xe^{3x^2+2}

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

\frac{1}{6} e^{3 x^{2} + 2} + C

Solution steps

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

Apply u-substitution

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

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

Apply u-substitution

: \int e^{v} \frac{1}{3} d v

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

Use the common integral:

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

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

Substitute back

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

Simplify

\frac{1}{2} \cdot \frac{1}{3} e^{3 x^{2} + 2} : \frac{1}{6} e^{3 x^{2} + 2}

= \frac{1}{6} e^{3 x^{2} + 2}

Add a constant to the solution

= \frac{1}{6} e^{3 x^{2} + 2} + C

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What is the integral of xe^{3x^2+2} ?
The integral of xe^{3x^2+2} is 1/6 e^{3x^2+2}+C