What is the integral of cos^2((2x)/3) ?

The integral of cos^2((2x)/3) is 1/2 x+3/8 sin((4x)/3)+C

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integral of cos^2((2x)/3)

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

\frac{1}{2} x + \frac{3}{8} sin (\frac{4 x}{3}) + C

Solution steps

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

Rewrite using trig identities

= \int 1 - sin^{2} (\frac{2 x}{3}) 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{2 x}{3}) d x

\int 1 d x = x

\int sin^{2} (\frac{2 x}{3}) d x = \frac{1}{2} (x - \frac{3}{4} sin (\frac{4 x}{3}))

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

Simplify

x - \frac{1}{2} (x - \frac{3}{4} sin (\frac{4 x}{3})) : \frac{1}{2} x + \frac{3}{8} sin (\frac{4 x}{3})

= \frac{1}{2} x + \frac{3}{8} sin (\frac{4 x}{3})

Add a constant to the solution

= \frac{1}{2} x + \frac{3}{8} sin (\frac{4 x}{3}) + C

n = 0 \sum \infty 2 + e^{- 3 n}

g (s) = \frac{5}{s - 3}

\frac{d}{d x} (- 2 x cos (x))

\frac{5}{2} y^{4} y^{^{'}} = \frac{y ^{5} + 4}{x ^{3}}

(2 x + y) d x + (x + 3 y) d y = 0

What is the integral of cos^2((2x)/3) ?
The integral of cos^2((2x)/3) is 1/2 x+3/8 sin((4x)/3)+C