What is the integral of x*e^{-2x^2} ?

The integral of x*e^{-2x^2} is -1/4 e^{-2x^2}+C

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integral of x*e^{-2x^2}

\int x \cdot e^{- 2 x^{2}} d x

- \frac{1}{4} e^{- 2 x^{2}} + C

Solution steps

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

Apply u-substitution

: \int - \frac{e ^{u}}{4} d u

= \int - \frac{e ^{u}}{4} d u

Take the constant out:

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

= - \frac{1}{4} \cdot \int e^{u} d u

Use the common integral:

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

= - \frac{1}{4} e^{u}

Substitute back

u = - 2 x^{2}

= - \frac{1}{4} e^{- 2 x^{2}}

Add a constant to the solution

= - \frac{1}{4} e^{- 2 x^{2}} + C

\frac{d}{d x} (a x + b)

\frac{d}{d x} (\frac{2 x ^{2}}{4 - x ^{2}})

\int_{2}^{4} (x - 9) d x

\int \frac{1}{x ^{2} + x ^{3}} d x

y^{^{'''}} - 2 y^{^{''}} - 4 y^{^{'}} + 8 y = 6 x e^{2 x}

What is the integral of x*e^{-2x^2} ?
The integral of x*e^{-2x^2} is -1/4 e^{-2x^2}+C