What is the integral of-x+3(cos((n*pi*x)/8)) ?

The integral of-x+3(cos((n*pi*x)/8)) is -(x^2)/2+(24}{pin}sin(\frac{pinx)/8)+C

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integral of-x+3(cos((npix)/8))

\int - x + 3 (cos (\frac{n \cdot π \cdot x}{8})) d x

- \frac{x ^{2}}{2} + \frac{24}{πn} sin (\frac{πn x}{8}) + C

Solution steps

\int - x + 3 cos (\frac{nπ x}{8}) 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 d x + \int 3 cos (\frac{nπ x}{8}) d x

\int x d x = \frac{x ^{2}}{2}

\int 3 cos (\frac{nπ x}{8}) d x = \frac{24}{πn} sin (\frac{πn x}{8})

= - \frac{x ^{2}}{2} + \frac{24}{πn} sin (\frac{πn x}{8})

Add a constant to the solution

= - \frac{x ^{2}}{2} + \frac{24}{πn} sin (\frac{πn x}{8}) + C

\frac{d}{d x} (\frac{1}{x y})

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

y = t

\int \frac{2 x - 1}{x ^{2}} d x

3 x^{3} - x^{2} + 9 x - 9

What is the integral of-x+3(cos((n*pi*x)/8)) ?
The integral of-x+3(cos((n*pi*x)/8)) is -(x^2)/2+(24}{pin}sin(\frac{pinx)/8)+C