What is the integral of (6x)/(e^{3x^2)} ?

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

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integral of (6x)/(e^{3x^2)}

\int \frac{6 x}{e ^{3 x^{2}}} d x

- e^{- 3 x^{2}} + C

Solution steps

\int \frac{6 x}{e ^{3 x^{2}}} d x

Take the constant out:

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

= 6 \cdot \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}}

= 6 \cdot \int x e^{- 3 x^{2}} d x

Apply u-substitution

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

= 6 \cdot \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

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

Use the common integral:

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

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

Substitute back

u = - 3 x^{2}

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

Simplify

6 (- \frac{1}{6} e^{- 3 x^{2}}) : - e^{- 3 x^{2}}

= - e^{- 3 x^{2}}

Add a constant to the solution

= - e^{- 3 x^{2}} + C

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