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

The integral of cos(2/3 x) is xcos(2/3 x)-xcos((2x)/3)+3/2 sin((2x)/3)+C

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integral of cos(2/3 x)

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

x cos (\frac{2}{3} x) - x cos (\frac{2 x}{3}) + \frac{3}{2} sin (\frac{2 x}{3}) + C

Solution steps

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

\frac{2}{3} = 2 \cdot \frac{1}{3}

= \int cos (2 \cdot \frac{1}{3} x) d x

Apply Integration By Parts

: x cos (\frac{2}{3} x) - \int - \frac{2}{3} x sin (\frac{2 x}{3}) d x

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

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

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

Simplify

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

= x cos (\frac{2}{3} x) - x cos (\frac{2 x}{3}) + \frac{3}{2} sin (\frac{2 x}{3})

Add a constant to the solution

= x cos (\frac{2}{3} x) - x cos (\frac{2 x}{3}) + \frac{3}{2} sin (\frac{2 x}{3}) + C

y^{^{''}} + 4 y^{^{'}} + 3 y = 20 t e^{5 t}

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

\int \frac{x ^{3}}{4 + x ^{4}} d x

\frac{d}{d x} (7 x + 9)

n = 0 \sum \infty \frac{1}{n + 6}

What is the integral of cos(2/3 x) ?
The integral of cos(2/3 x) is xcos(2/3 x)-xcos((2x)/3)+3/2 sin((2x)/3)+C