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

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

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integral of cos^2(1/2 x)

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

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

Solution steps

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

Rewrite using trig identities

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

\int 1 d x = x

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

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

Simplify

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

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

Add a constant to the solution

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

\int \frac{x ^{2} - 5 x + 6}{x ^{2} + 4} d x

\frac{\partial}{\partial x} ((\frac{y}{x})^{\frac{3}{4}})

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

y^{^{'}} + 4 y = 32 x, y (0) = 3

\frac{d y}{d x} = x y + 5 x + 4 y + 20

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