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

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

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

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

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

Solution steps

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

Simplify

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

Rewrite using trig identities

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

Take the constant out:

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

= \frac{1}{2} \cdot \int 1 + cos (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

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

\int 1 d x = x

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

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

Add a constant to the solution

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

f (x) = 3 x^{3} - x^{2} + 1, (2, 21)

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

\int (2 x)^{2} d x

(- 2, - 6), (2, 2)

f (x) = 9 + 4 x^{2} - 2 x^{3}, at x = 1

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