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

The integral of sqrt(cos^2(2x)) is 1/2 sin(2x)+C

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integral of sqrt(cos^2(2x))

\int cos^{2} (2 x) d x

\frac{1}{2} sin (2 x) + C

Solution steps

\int cos^{2} (2 x) d x

cos^{2} (2 x) = (cos (2 x)),

assuming

cos (2 x) \geq 0

= \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 common integral:

\int cos (u) d u = sin (u)

= \frac{1}{2} sin (u)

Substitute back

u = 2 x

= \frac{1}{2} sin (2 x)

Add a constant to the solution

= \frac{1}{2} sin (2 x) + C

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What is the integral of sqrt(cos^2(2x)) ?
The integral of sqrt(cos^2(2x)) is 1/2 sin(2x)+C