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

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

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

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

4 e \frac{π}{2} erf (x + \frac{1}{2}) + C

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

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

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

Take the constant out:

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

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

Use the nonelementary integral:

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

= 4 e \frac{π}{2} erf (u)

Substitute back

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

= 4 e \frac{π}{2} erf (x + \frac{1}{2})

Add a constant to the solution

= 4 e \frac{π}{2} erf (x + \frac{1}{2}) + C

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