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

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

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integral of cos((npix)/3)

\int cos (\frac{nπ x}{3}) d x

\frac{3}{πn} sin (\frac{πn x}{3}) + C

Solution steps

\int cos (\frac{nπ x}{3}) d x

Apply Integration By Parts

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

= 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})

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

Simplify

= \frac{3}{πn} sin (\frac{πn x}{3})

Add a constant to the solution

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

(x^{2} + 1) d x + \frac{d y}{y} = 0, y (- 1) = 1

\frac{( 3 + 2 x )}{( 1 + x )}

\int cos^{2} (θ) sin (θ) d θ

\int x e^{(x^{2})} d x

\int_{- 1}^{1} \frac{2 x}{( x + 5 ) ^{2}} d x

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