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

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

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

integral

e^{2 x - x^{2}}

e \frac{π}{2} erf (x - 1) + C

Solution steps

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

Complete the square

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

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

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 = x - 1

= e \frac{π}{2} erf (x - 1)

Add a constant to the solution

= e \frac{π}{2} erf (x - 1) + C

in t e g r a l \frac{1}{( 2 ^{t} - 1 )}

in t e g r a l 4 x^{2} + 3 x + 2 \cdot 2 (x + 1)

in t e g r a l cos^{2} (\frac{x}{4})

in t e g r a l (\frac{1}{x ^{4}}) sin (x)

in t e g r a l cos ((\frac{1}{4}) x)

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