What is the integral of sqrt(1-cos(x)) ?

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

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integral of sqrt(1-cos(x))

\int 1 - cos (x) d x

- 22 cos (\frac{x}{2}) + C

Solution steps

\int 1 - cos (x) d x

Rewrite using trig identities

= \int 2 sin^{2} (\frac{x}{2}) d x

Take the constant out:

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

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

sin^{2} (\frac{x}{2}) = (sin (\frac{x}{2})),

assuming

sin (\frac{x}{2}) \geq 0

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

Apply u-substitution

: \int sin (u) \cdot 2 d u

= 2 \cdot \int sin (u) \cdot 2 d u

Take the constant out:

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

= 2 \cdot 2 \cdot \int sin (u) d u

Use the common integral:

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

= 2 \cdot 2 (- cos (u))

Substitute back

u = \frac{x}{2}

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

Simplify

= - 22 cos (\frac{x}{2})

Add a constant to the solution

= - 22 cos (\frac{x}{2}) + C

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