What is the integral of (4x)/(e^{5x^2)} ?

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

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integral of (4x)/(e^{5x^2)}

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

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

Solution steps

\int \frac{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 \frac{x}{e ^{5 x^{2}}} d x

Apply exponent rule:

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

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

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

Apply u-substitution

= 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

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