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

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

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

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

- e^{- 2 x} + C

Solution steps

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

Take the constant out:

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

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

Apply exponent rule:

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

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

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

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = - 2 x

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

Simplify

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

= - e^{- 2 x}

Add a constant to the solution

= - e^{- 2 x} + C