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

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

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

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

2 sin (\frac{x}{2}) + C

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

: \int 2 cos (u) d u

= \int 2 cos (u) d u

Take the constant out:

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

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

Use the common integral:

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

= 2 sin (u)

Substitute back

u = \frac{x}{2}

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

Add a constant to the solution

= 2 sin (\frac{x}{2}) + C

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\int t^{2} \cdot e^{- s t} d t

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