What is the integral from-a to a of e^{(-x^2)/(a^2)} ?

The integral from-a to a of e^{(-x^2)/(a^2)} is a*1.49364…

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integral from-a to a of e^{(-x^2)/(a^2)}

\int_{- a}^{a} e^{\frac{- x ^{2}}{a ^{2}}} d x

a \cdot 1.49364 \dots

Solution steps

\int_{- a}^{a} e^{\frac{- x ^{2}}{a ^{2}}} d x

Simplify

= \int_{- a}^{a} e^{- \frac{x ^{2}}{a ^{2}}} d x

Simplify

\frac{x ^{2}}{a ^{2}} : (\frac{x}{a})^{2}

= \int_{- a}^{a} e^{- (\frac{x}{a})^{2}} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Use the nonelementary integral:

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

= a [\frac{π}{2} erf (u)]_{- 1}^{1}

Compute the boundaries:

1.49364 \dots

= a \cdot 1.49364 \dots

(- 4, 3), (3, 1)

\int \frac{x ^{2} - 15}{x + 7} d x

y^{^{''}} - 2 y^{^{'}} = 9 x

\int_{0}^{2} (3 x^{2} - 2) d x

(2 + u^{2})^{5} (5 - 9 u^{2})^{8}

What is the integral from-a to a of e^{(-x^2)/(a^2)} ?
The integral from-a to a of e^{(-x^2)/(a^2)} is a*1.49364…