What is the integral of sqrt(cos(2x)) ?

The integral of sqrt(cos(2x)) is \E(2\vert (2x)/2)+C

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integral of sqrt(cos(2x))

\int cos (2 x) d x

E (2∣ \frac{2 x}{2}) + C

Solution steps

\int cos (2 x) d x

Apply u-substitution

: \int cos (u) \frac{1}{2} d u

= \int 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 cos (u) d u

Use the nonelementary integral:

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

= \frac{1}{2} \cdot 2 E (\frac{u}{2} ∣ 2)

Substitute back

u = 2 x

= \frac{1}{2} \cdot 2 E (\frac{2 x}{2} ∣ 2)

Simplify

\frac{1}{2} \cdot 2 E (\frac{2 x}{2} ∣ 2) : E (2∣ \frac{2 x}{2})

= E (2∣ \frac{2 x}{2})

Add a constant to the solution

= E (2∣ \frac{2 x}{2}) + C

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What is the integral of sqrt(cos(2x)) ?
The integral of sqrt(cos(2x)) is \E(2\vert (2x)/2)+C