What is the integral of sqrt(2-2cos(4x)) ?

The integral of sqrt(2-2cos(4x)) is -cos(2x)+C

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integral of sqrt(2-2cos(4x))

\int 2 - 2 cos (4 x) d x

- cos (2 x) + C

Solution steps

\int 2 - 2 cos (4 x) d x

2 - 2 cos (4 x) = 2 1 - cos (4 x)

= \int 2 1 - cos (4 x) d x

Take the constant out:

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

= 2 \cdot \int 1 - cos (4 x) d x

Apply Integral Substitution

= 2 \cdot \int \frac{1}{2} 1 - cos (2 u) d u

Take the constant out:

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

= 2 \frac{1}{2} \cdot \int 1 - cos (2 u) d u

Rewrite using trig identities

= 2 \frac{1}{2} \cdot \int 2 sin^{2} (u) d u

Take the constant out:

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

= 2 \frac{1}{2} 2 \cdot \int sin^{2} (u) d u

sin^{2} (u) = (sin (u)),

assuming

sin (u) \geq 0

= 2 \frac{1}{2} 2 \cdot \int sin (u) d u

Use the common integral:

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

= 2 \frac{1}{2} 2 (- cos (u))

Substitute back

u = 2 x

= 2 \frac{1}{2} 2 (- cos (2 x))

Simplify

2 \frac{1}{2} 2 (- cos (2 x)) : - cos (2 x)

= - cos (2 x)

Add a constant to the solution

= - cos (2 x) + C

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What is the integral of sqrt(2-2cos(4x)) ?
The integral of sqrt(2-2cos(4x)) is -cos(2x)+C