What is the integral from 0 to pi/4 of sqrt(cos(2x)) ?

The integral from 0 to pi/4 of sqrt(cos(2x)) is \E(2*(pi/2)/2 \vert)-\E(0*2\vert)

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integral from 0 to pi/4 of sqrt(cos(2x))

\int_{0}^{\frac{π}{4}} cos (2 x) d x

E (2 \cdot \frac{\frac{π}{2}}{2} ∣) - E (0 \cdot 2∣)

Solution steps

\int_{0}^{\frac{π}{4}} cos (2 x) d x

Apply u-substitution

: \int_{0}^{\frac{π}{2}} cos (u) \frac{1}{2} d u

= \int_{0}^{\frac{π}{2}} cos (u) \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_{0}^{\frac{π}{2}} cos (u) d u

Use the nonelementary integral:

\int cos (x) d x = 2 E (\frac{x}{2} ∣ 2)

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

Compute the boundaries:

2 E (\frac{\frac{π}{2}}{2} ∣ 2) - 2 E (0 \cdot 2∣)

= \frac{1}{2} (2 E (\frac{\frac{π}{2}}{2} ∣ 2) - 2 E (0 \cdot 2∣))

Simplify

= E (2 \cdot \frac{\frac{π}{2}}{2} ∣) - E (0 \cdot 2∣)

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What is the integral from 0 to pi/4 of sqrt(cos(2x)) ?
The integral from 0 to pi/4 of sqrt(cos(2x)) is \E(2*(pi/2)/2 \vert)-\E(0*2\vert)