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

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

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integral of sqrt(e^{-2x)}

\int e^{- 2 x} d x

- e^{- x} + C

Solution steps

\int e^{- 2 x} d x

Apply u-substitution

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

Take the constant out:

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

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

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

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

= - \frac{1}{2} \cdot 2 e^{v}

Substitute back

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

Simplify

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

= - e^{- x}

Add a constant to the solution

= - e^{- x} + C

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