What is the integral of 4xe^{-5x^2} ?

The integral of 4xe^{-5x^2} is -2/5 e^{-5x^2}+C

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integral of 4xe^{-5x^2}

\int 4 x e^{- 5 x^{2}} d x

- \frac{2}{5} e^{- 5 x^{2}} + C

Solution steps

\int 4 x e^{- 5 x^{2}} d x

Take the constant out:

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

= 4 \cdot \int x e^{- 5 x^{2}} d x

Apply u-substitution

: \int - \frac{e ^{u}}{10} d u

= 4 \cdot \int - \frac{e ^{u}}{10} d u

Take the constant out:

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

= 4 (- \frac{1}{10} \cdot \int e^{u} d u)

Use the common integral:

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

= 4 (- \frac{1}{10} e^{u})

Substitute back

u = - 5 x^{2}

= 4 (- \frac{1}{10} e^{- 5 x^{2}})

Simplify

4 (- \frac{1}{10} e^{- 5 x^{2}}) : - \frac{2}{5} e^{- 5 x^{2}}

= - \frac{2}{5} e^{- 5 x^{2}}

Add a constant to the solution

= - \frac{2}{5} e^{- 5 x^{2}} + C

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What is the integral of 4xe^{-5x^2} ?
The integral of 4xe^{-5x^2} is -2/5 e^{-5x^2}+C