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

The integral from 0 to 1 of e^{-(2x-1)^2} is 0.74682…

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

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

0.74682 \dots

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the nonelementary integral:

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

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

Compute the boundaries:

1.49364 \dots

= \frac{1}{2} \cdot 1.49364 \dots

Simplify

= 0.74682 \dots

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What is the integral from 0 to 1 of e^{-(2x-1)^2} ?
The integral from 0 to 1 of e^{-(2x-1)^2} is 0.74682…