What is the integral from 0 to 1/2 of cosh(x^2) ?

The integral from 0 to 1/2 of cosh(x^2) is 0.5(0.46128…+0.88622…\erfi(1/2))

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integral from 0 to 1/2 of cosh(x^2)

\int_{0}^{\frac{1}{2}} cosh (x^{2}) d x

0.5 (0.46128 \dots + 0.88622 \dots erfi (\frac{1}{2}))

Solution steps

\int_{0}^{\frac{1}{2}} cosh (x^{2}) d x

Rewrite using trig identities

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

Take the constant out:

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

= \frac{1}{2} \cdot \int_{0}^{\frac{1}{2}} e^{- x^{2}} + e^{x^{2}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int_{0}^{\frac{1}{2}} e^{- x^{2}} d x = 0.46128 \dots

\int_{0}^{\frac{1}{2}} e^{x^{2}} d x = \frac{π}{2} erfi (\frac{1}{2})

= \frac{1}{2} (0.46128 \dots + \frac{π}{2} erfi (\frac{1}{2}))

Simplify

= 0.5 (0.46128 \dots + 0.88622 \dots erfi (\frac{1}{2}))

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What is the integral from 0 to 1/2 of cosh(x^2) ?
The integral from 0 to 1/2 of cosh(x^2) is 0.5(0.46128…+0.88622…\erfi(1/2))