What is the integral of cos(2/5 x) ?

The integral of cos(2/5 x) is xcos(2/5 x)-xcos((2x)/5)+5/2 sin((2x)/5)+C

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integral of cos(2/5 x)

\int cos (\frac{2}{5} x) d x

x cos (\frac{2}{5} x) - x cos (\frac{2 x}{5}) + \frac{5}{2} sin (\frac{2 x}{5}) + C

Solution steps

\int cos (\frac{2}{5} x) d x

\frac{2}{5} = 2 \cdot \frac{1}{5}

= \int cos (2 \cdot \frac{1}{5} x) d x

Apply Integration By Parts

: x cos (\frac{2}{5} x) - \int - \frac{2}{5} x sin (\frac{2 x}{5}) d x

= x cos (\frac{2}{5} x) - \int - \frac{2}{5} x sin (\frac{2 x}{5}) d x

\int - \frac{2}{5} x sin (\frac{2 x}{5}) d x = - \frac{1}{2} (- 2 x cos (\frac{2 x}{5}) + 5 sin (\frac{2 x}{5}))

= x cos (\frac{2}{5} x) - (- \frac{1}{2} (- 2 x cos (\frac{2 x}{5}) + 5 sin (\frac{2 x}{5})))

Simplify

x cos (\frac{2}{5} x) - (- \frac{1}{2} (- 2 x cos (\frac{2 x}{5}) + 5 sin (\frac{2 x}{5}))) : x cos (\frac{2}{5} x) - x cos (\frac{2 x}{5}) + \frac{5}{2} sin (\frac{2 x}{5})

= x cos (\frac{2}{5} x) - x cos (\frac{2 x}{5}) + \frac{5}{2} sin (\frac{2 x}{5})

Add a constant to the solution

= x cos (\frac{2}{5} x) - x cos (\frac{2 x}{5}) + \frac{5}{2} sin (\frac{2 x}{5}) + C

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What is the integral of cos(2/5 x) ?
The integral of cos(2/5 x) is xcos(2/5 x)-xcos((2x)/5)+5/2 sin((2x)/5)+C