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

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

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

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

e \frac{π}{2} erf (x - 1) + C

Solution steps

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

Complete the square

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

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