What is the integral of cos((npix)/5) ?

The integral of cos((npix)/5) is 5/(pin)sin((pinx)/5)+C

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integral of cos((npix)/5)

\int cos (\frac{nπ x}{5}) d x

\frac{5}{πn} sin (\frac{πn x}{5}) + C

Solution steps

\int cos (\frac{nπ x}{5}) d x

Apply Integration By Parts

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

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

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

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

Simplify

= \frac{5}{πn} sin (\frac{πn x}{5})

Add a constant to the solution

= \frac{5}{πn} sin (\frac{πn x}{5}) + C

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What is the integral of cos((npix)/5) ?
The integral of cos((npix)/5) is 5/(pin)sin((pinx)/5)+C