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

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

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integral of (2x)/(e^{x^2+2)}

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

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

Solution steps

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

Take the constant out:

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

= 2 \cdot \int \frac{x}{e ^{x^{2} + 2}} d x

\frac{x}{e ^{x^{2} + 2}} = \frac{1}{e ^{2}} \frac{x}{e ^{x^{2}}}

= 2 \cdot \int \frac{1}{e ^{2}} \cdot \frac{x}{e ^{x^{2}}} d x

Take the constant out:

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

= 2 \cdot \frac{1}{e ^{2}} \cdot \int \frac{x}{e ^{x^{2}}} d x

Apply exponent rule:

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

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

= 2 \cdot \frac{1}{e ^{2}} \cdot \int x e^{- x^{2}} d x

Apply u-substitution

: \int - \frac{e ^{u}}{2} d u

= 2 \cdot \frac{1}{e ^{2}} \cdot \int - \frac{e ^{u}}{2} d u

Take the constant out:

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

= 2 \cdot \frac{1}{e ^{2}} (- \frac{1}{2} \cdot \int e^{u} d u)

Use the common integral:

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

= 2 \cdot \frac{1}{e ^{2}} (- \frac{1}{2} e^{u})

Substitute back

u = - x^{2}

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

Simplify

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

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

Add a constant to the solution

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

\frac{d}{d x} (\frac{128}{x})

x \to 2 + lim (\frac{4}{( x - 2 ) ^{5}})

n = 1 \sum \infty \frac{7}{n ^{2} + 81}

f (x) = - x^{2} + 25

\int (x + 1)^{2} d x

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