What is the integral of (x+4)cos(npix) ?

The integral of (x+4)cos(npix) is 1/(pi^2n^2)(pinxsin(pinx)+cos(pinx))+4/(pin)sin(pinx)+C

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integral of (x+4)cos(npix)

\int (x + 4) cos (nπ x) d x

\frac{1}{π ^{2} n ^{2}} (πn x sin (πn x) + cos (πn x)) + \frac{4}{πn} sin (πn x) + C

Solution steps

\int (x + 4) cos (nπ x) d x

Expand

(x + 4) cos (nπ x) : x cos (πn x) + 4 cos (πn x)

= \int x cos (πn x) + 4 cos (πn x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x cos (πn x) d x + \int 4 cos (πn x) d x

\int x cos (πn x) d x = \frac{1}{π ^{2} n ^{2}} (πn x sin (πn x) + cos (πn x))

\int 4 cos (πn x) d x = \frac{4}{πn} sin (πn x)

= \frac{1}{π ^{2} n ^{2}} (πn x sin (πn x) + cos (πn x)) + \frac{4}{πn} sin (πn x)

Add a constant to the solution

= \frac{1}{π ^{2} n ^{2}} (πn x sin (πn x) + cos (πn x)) + \frac{4}{πn} sin (πn x) + C

\int_{1}^{4} \frac{1}{w ^{2}} d w

\int (x + 2)^{2} (1 - x)^{6} d x

5 - x^{2}, sin (x), 0, 2

\frac{d ^{4}}{d x ^{4}} (8 e^{- 2 x})

\frac{d y}{d x} = (x - 2) e^{2 y}

What is the integral of (x+4)cos(npix) ?
The integral of (x+4)cos(npix) is 1/(pi^2n^2)(pinxsin(pinx)+cos(pinx))+4/(pin)sin(pinx)+C