What is the integral of e^{2y-y^2} ?

The integral of e^{2y-y^2} is e(sqrt(pi))/2 \erf(y-1)+C

integral of e^{2y-y^2}

\int e^{2 y - y^{2}} d y

e \frac{π}{2} erf (y - 1) + C

Solution steps

\int e^{2 y - y^{2}} d y

Complete the square

2 y - y^{2} : - (y - 1)^{2} + 1

= \int e^{- (y - 1)^{2} + 1} d y

Apply u-substitution

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

e^{- u^{2} + 1} = e^{1} e^{- u^{2}}

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

Take the constant out:

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

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

Simplify

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

Use the nonelementary integral:

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

= e \frac{π}{2} erf (u)

Substitute back

u = y - 1

= e \frac{π}{2} erf (y - 1)

Add a constant to the solution

= e \frac{π}{2} erf (y - 1) + C

3 \frac{d y}{d x} = \frac{e ^{y} sin ^{2} ( x )}{y sec ( x )}

\int \frac{x + 1}{( x ^{2} + 4 x + 5 ) ^{2}} d x

d er i v a t i v e f (x) = (4 a + 1)^{2}

s l o p e y = \frac{x ln ( x )}{6} + \frac{2 x}{3}

\int 5 x e^{2 x} d x

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