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

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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

= \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

\frac{x}{( x y + 1 ) ^{2}}

\frac{1}{( 2 ^{t} - 1 )}

\frac{( t - 1 ) ^{2}}{( 2 t + 1 ) ^{2}}

\frac{p}{( p ^{2} + 4 )}

4 x^{2} + 3 x + 2 \cdot 2 (x + 1)

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