What is the integral from 0 to 1 of e^{x+x^2} ?

The integral from 0 to 1 of e^{x+x^2} is (\sqrt[4]{e}(sqrt(pi)\erfi(3/2)-sqrt(pi)\erfi(1/2)))/2

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integral from 0 to 1 of e^{x+x^2}

\int_{0}^{1} e^{x + x^{2}} d x

Solution steps

\int_{0}^{1} e^{x + x^{2}} d x

Complete the square

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

= \int_{0}^{1} e^{(x + \frac{1}{2})^{2} - \frac{1}{4}} d x

Apply u-substitution

= \int_{\frac{1}{2}}^{\frac{3}{2}} 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)

Compute the boundaries:

\frac{π erfi ( \frac{3}{2} ) - π erfi ( \frac{1}{2} )}{2}

Simplify

What is the integral from 0 to 1 of e^{x+x^2} ?
The integral from 0 to 1 of e^{x+x^2} is (\sqrt[4]{e}(sqrt(pi)\erfi(3/2)-sqrt(pi)\erfi(1/2)))/2