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

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

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integral of sqrt(1-cos(θ))

\int 1 - cos (θ) d θ

- 22 cos (\frac{θ}{2}) + C

Solution steps

\int 1 - cos (θ) d θ

Rewrite using trig identities

= \int 2 sin^{2} (\frac{θ}{2}) d θ

Take the constant out:

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

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

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

assuming

sin (\frac{θ}{2}) \geq 0

= 2 \cdot \int sin (\frac{θ}{2}) d θ

Apply u-substitution

= 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{θ}{2}

= 2 \cdot 2 (- cos (\frac{θ}{2}))

Simplify

= - 22 cos (\frac{θ}{2})

Add a constant to the solution

= - 22 cos (\frac{θ}{2}) + C

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