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

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

integral of e^{-sqrt(y)}

\int e^{- y} d y

2 (- e^{- y} y - e^{- y}) + C

Solution steps

\int e^{- y} d y

Apply u-substitution

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

Apply Integration By Parts

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

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

= 2 (e^{u} u - e^{u})

Substitute back

u = - y

= 2 (e^{- y} (- y) - e^{- y})

Simplify

= 2 (- e^{- y} y - e^{- y})

Add a constant to the solution

= 2 (- e^{- y} y - e^{- y}) + C

\frac{d}{d x} (ln (3 x + 1))

x \to 0 lim (\frac{1 - 2 cos ( x ) + cos ( 2 x )}{x ^{2}})

d er i v a t i v e y = - \frac{2}{5} x - 3

\frac{d}{d x} (e^{c o s (2 x)} sin (2 x))

x \to \infty lim (\frac{2 e ^{2 x}}{2 x})

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