What is the integral of cos^2(x/4) ?

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

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integral of cos^2(x/4)

integral

cos^{2} (\frac{x}{4})

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

Solution steps

\int cos^{2} (\frac{x}{4}) d x

Rewrite using trig identities

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

\int 1 d x = x

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

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

Simplify

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

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

Add a constant to the solution

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

in t e g r a l (\frac{1}{x ^{4}}) sin (x)

in t e g r a l cos ((\frac{1}{4}) x)

in t e g r a l \frac{( 1 + x ) ^{3}}{x}

in t e g r a l \frac{u}{( 1 + u ^{2} )}

in t e g r a l \frac{2 tan ( x )}{( 1 + 2 tan ^{2} ( x ) )}

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