What is the integral of 3cos((npix)/3) ?

The integral of 3cos((npix)/3) is 9/(pin)sin((pinx)/3)+C

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integral of 3cos((npix)/3)

\int 3 cos (\frac{nπ x}{3}) d x

\frac{9}{πn} sin (\frac{πn x}{3}) + C

Solution steps

\int 3 cos (\frac{nπ x}{3}) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

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

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

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

Simplify

3 (x cos (\frac{πn x}{3}) - (x cos (\frac{πn x}{3}) - \frac{3}{πn} sin (\frac{πn x}{3}))) : \frac{9}{πn} sin (\frac{πn x}{3})

= \frac{9}{πn} sin (\frac{πn x}{3})

Add a constant to the solution

= \frac{9}{πn} sin (\frac{πn x}{3}) + C

y^{^{'}} = \frac{x ^{2}}{y ( 2 + x ^{3} )}

\frac{\partial}{\partial x} (3 e^{x} cos (y))

\frac{d}{d x} ((x - 4)^{2} x^{2})

\frac{\partial}{\partial x} (- \frac{9 x ^{2}}{x ^{2} + y ^{2}})

\int_{4}^{4} e^{- x^{2}} d x

What is the integral of 3cos((npix)/3) ?
The integral of 3cos((npix)/3) is 9/(pin)sin((pinx)/3)+C