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

The integral of e^{-2sqrt(x)} is -e^{-2sqrt(x)}sqrt(x)-1/2 e^{-2sqrt(x)}+C

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

\int e^{- 2 x} d x

- e^{- 2 x} x - \frac{1}{2} e^{- 2 x} + C

Solution steps

\int e^{- 2 x} d x

Apply u-substitution

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

Take the constant out:

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

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

Apply u-substitution

= 2 \cdot \int \frac{e ^{v} v}{4} d v

Take the constant out:

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

= 2 \cdot \frac{1}{4} \cdot \int e^{v} v d v

Apply Integration By Parts

= 2 \cdot \frac{1}{4} (e^{v} v - \int e^{v} d v)

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

= 2 \cdot \frac{1}{4} (e^{v} v - e^{v})

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

What is the integral of e^{-2sqrt(x)} ?
The integral of e^{-2sqrt(x)} is -e^{-2sqrt(x)}sqrt(x)-1/2 e^{-2sqrt(x)}+C