What is the integral of e^{-x^2+3x} ?

The integral of e^{-x^2+3x} is e^2\sqrt[4]{e}(sqrt(pi))/2 \erf(x-3/2)+C

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integral of e^{-x^2+3x}

\int e^{- x^{2} + 3 x} d x

Solution steps

\int e^{- x^{2} + 3 x} d x

Complete the square

- x^{2} + 3 x : - (x - \frac{3}{2})^{2} + \frac{9}{4}

= \int e^{- (x - \frac{3}{2})^{2} + \frac{9}{4}} d x

Apply u-substitution

: \int e^{- u^{2} + \frac{9}{4}} d u

= \int e^{- u^{2} + \frac{9}{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} erf (u)

Substitute back

u = x - \frac{3}{2}

Add a constant to the solution

e^{- 5 t}

\frac{d y}{d x} = 10 x

\int \frac{1}{csc ^{4} ( 3 t ) sec ^{2} ( 3 t )} d t

\frac{\partial}{\partial y} (x^{3} - 12 x y + 8 y^{3})

\frac{\partial}{\partial x} (\frac{e ^{y}}{e ^{x} + e ^{y}})

What is the integral of e^{-x^2+3x} ?
The integral of e^{-x^2+3x} is e^2\sqrt[4]{e}(sqrt(pi))/2 \erf(x-3/2)+C