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

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

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

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

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

Solution steps

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

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{x}{e ^{3 x^{2}}} = (x) e^{- 3 x^{2}}

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

Apply u-substitution

= \int - \frac{e ^{u}}{6} d u

Take the constant out:

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

= - \frac{1}{6} \cdot \int e^{u} d u

Use the common integral:

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

= - \frac{1}{6} e^{u}

Substitute back

u = - 3 x^{2}

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

Add a constant to the solution

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

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