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

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

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integral of sqrt((1-cos(x))/2)

\int \frac{1 - cos ( x )}{2} d x

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

Solution steps

\int \frac{1 - cos ( x )}{2} d x

Apply u-substitution

: \int \frac{2 u}{( 1 + u ^{2} ) 1 + u ^{2}} d u

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

Apply u-substitution

: \int \frac{1}{2 v v} d v

= 2 \cdot \int \frac{1}{2 v v} d v

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{v v} d v

Simplify

v v : v^{\frac{3}{2}}

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{v ^{\frac{3}{2}}} d v

Apply the Power Rule

: - \frac{2}{v}

= 2 \cdot \frac{1}{2} (- \frac{2}{v})

Substitute back

= 2 \cdot \frac{1}{2} - \frac{2}{1 + tan ^{2} ( \frac{x}{2} )}

Simplify

2 \cdot \frac{1}{2} - \frac{2}{1 + tan ^{2} ( \frac{x}{2} )} : - 2 cos (\frac{x}{2})

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

Add a constant to the solution

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

\frac{\partial}{\partial t} (e^{- 2 t} cos (x))

x \to 0 lim (sin (\frac{7}{x}))

f (x) = \frac{9}{x}, at x = 1

\frac{d}{d t} (f (t) (t)^{- 1})

\int 18 cos (2 t) d t

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