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

The integral of (2-sin(x/2))^2 is 3/2 x+1/4 sin(2x)+2sin(x)+C

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integral of (2-sin(x/2))^2

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

\frac{3}{2} x + \frac{1}{4} sin (2 x) + 2 sin (x) + C

Solution steps

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

Rewrite using trig identities

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

Expand

(cos (x) + 1)^{2} : cos^{2} (x) + 2 cos (x) + 1

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

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

\int 2 cos (x) d x = 2 sin (x)

\int 1 d x = x

= \frac{1}{2} (x + \frac{1}{2} sin (2 x)) + 2 sin (x) + x

Simplify

\frac{1}{2} (x + \frac{1}{2} sin (2 x)) + 2 sin (x) + x : \frac{3}{2} x + \frac{1}{4} sin (2 x) + 2 sin (x)

= \frac{3}{2} x + \frac{1}{4} sin (2 x) + 2 sin (x)

Add a constant to the solution

= \frac{3}{2} x + \frac{1}{4} sin (2 x) + 2 sin (x) + C

\int \frac{ln ( x ^{13} )}{x} d x

\int \frac{( 3 x - 7 x ^{2} )}{x} d x

\int (\frac{2}{x + x ^{3} - 5 x}) d x

What is the integral of (2-sin(x/2))^2 ?
The integral of (2-sin(x/2))^2 is 3/2 x+1/4 sin(2x)+2sin(x)+C