What is the integral from 0 to 1 of e^{-5x^2} ?

The integral from 0 to 1 of e^{-5x^2} is 0.39571…

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integral from 0 to 1 of e^{-5x^2}

\int_{0}^{1} e^{- 5 x^{2}} d x

0.39571 \dots

Solution steps

\int_{0}^{1} e^{- 5 x^{2}} d x

= \int_{0}^{1} e^{- (5 x)^{2}} d x

Apply u-substitution

: \int_{0}^{5} e^{- u^{2}} \frac{1}{5} d u

= \int_{0}^{5} e^{- u^{2}} \frac{1}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int_{0}^{5} e^{- u^{2}} d u

Use the nonelementary integral:

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

= \frac{1}{5} [\frac{π}{2} erf (u)]_{0}^{5}

Compute the boundaries:

0.88483 \dots

= \frac{1}{5} \cdot 0.88483 \dots

Simplify

= 0.39571 \dots

x \to \infty lim (e^{\frac{1}{2} - \frac{1}{x ^{2}}} - 1)

n = 1 \sum \infty 1 (\frac{1}{6})^{n}

n = 1 \sum \infty n^{n} (x - 3)^{n}

\int_{0}^{\frac{π}{2}} \frac{cos ( θ )}{1 + sin ( θ )} d θ

n = 0 \sum \infty 2 (\frac{- 2}{3})^{n}

What is the integral from 0 to 1 of e^{-5x^2} ?
The integral from 0 to 1 of e^{-5x^2} is 0.39571…