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 \erfi(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} erfi (u)

Substitute back

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

Add a constant to the solution

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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 \erfi(x+3/2)+C