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

The integral of e^{(x^2)-x} is \sqrt[4]{e}(sqrt(pi))/2 \erfi(x-1/2)+C

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

integral

e^{(x^{2}) - x}

Solution steps

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

Complete the square

x^{2} - x : (x - \frac{1}{2})^{2} - \frac{1}{4}

= \int e^{(x - \frac{1}{2})^{2} - \frac{1}{4}} d x

Apply u-substitution

: \int e^{u^{2} - \frac{1}{4}} d u

= \int e^{u^{2} - \frac{1}{4}} d u

Simplify

Take the constant out:

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

Use the nonelementary integral:

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

Substitute back

u = x - \frac{1}{2}

Add a constant to the solution

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

\frac{sin ^{2} ( 2 x )}{cos ^{2} ( x )}

e^{\frac{- x ^{2}}{3}}

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

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

What is the integral of e^{(x^2)-x} ?
The integral of e^{(x^2)-x} is \sqrt[4]{e}(sqrt(pi))/2 \erfi(x-1/2)+C