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

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

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integral of sqrt(1-cos(u))

\int 1 - cos (u) d u

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

Solution steps

\int 1 - cos (u) d u

Rewrite using trig identities

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

Take the constant out:

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

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

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

assuming

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

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

v = \frac{u}{2}

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

Simplify

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

Add a constant to the solution

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

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\frac{1}{1 - x ^{2}}

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