What is the integral from-1 to 1 of e^{3x^2} ?

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

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integral from-1 to 1 of e^{3x^2}

\int_{- 1}^{1} e^{3 x^{2}} d x

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

Solution steps

\int_{- 1}^{1} e^{3 x^{2}} d x

= \int_{- 1}^{1} e^{(3 x)^{2}} d x

Apply u-substitution

: \int_{- 3}^{3} e^{u^{2}} \frac{1}{3} d u

= \int_{- 3}^{3} e^{u^{2}} \frac{1}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int_{- 3}^{3} e^{u^{2}} d u

Use the nonelementary integral:

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

= \frac{1}{3} [\frac{π}{2} erfi (u)]_{- 3}^{3}

Compute the boundaries:

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

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

Simplify

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

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