What is the integral of (cos(x/2)+sin(x/2))^2 ?

The integral of (cos(x/2)+sin(x/2))^2 is x-cos(x)+C

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integral of (cos(x/2)+sin(x/2))^2

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

x - cos (x) + C

Solution steps

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

Rewrite using trig identities

= \int 1 + sin (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d x + \int sin (x) d x

\int 1 d x = x

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

= x - cos (x)

Add a constant to the solution

= x - cos (x) + C

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