What is the integral of 3xe^{(-3x^2)/4} ?

The integral of 3xe^{(-3x^2)/4} is -2e^{-(3x^2)/4}+C

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integral of 3xe^{(-3x^2)/4}

\int 3 x e^{\frac{- 3 x ^{2}}{4}} d x

- 2 e^{- \frac{3 x ^{2}}{4}} + C

Solution steps

\int 3 x e^{\frac{- 3 x ^{2}}{4}} d x

Take the constant out:

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

= 3 \cdot \int x e^{\frac{- 3 x ^{2}}{4}} d x

Simplify

= 3 \cdot \int e^{- \frac{3 x ^{2}}{4}} x d x

Apply u-substitution

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

Take the constant out:

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

= 3 (- \frac{2}{3} \cdot \int e^{u} d u)

Use the common integral:

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

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

Substitute back

u = - \frac{3 x ^{2}}{4}

= 3 (- \frac{2}{3} e^{- \frac{3 x ^{2}}{4}})

Simplify

3 (- \frac{2}{3} e^{- \frac{3 x ^{2}}{4}}) : - 2 e^{- \frac{3 x ^{2}}{4}}

= - 2 e^{- \frac{3 x ^{2}}{4}}

Add a constant to the solution

= - 2 e^{- \frac{3 x ^{2}}{4}} + C

What is the integral of 3xe^{(-3x^2)/4} ?
The integral of 3xe^{(-3x^2)/4} is -2e^{-(3x^2)/4}+C