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

The integral of (x+4)cos(n)pix is cos(n)pi((x^3)/3+2x^2)+C

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integral of (x+4)cos(n)pix

\int (x + 4) cos (n) π x d x

cos (n) π (\frac{x ^{3}}{3} + 2 x^{2}) + C

Solution steps

\int (x + 4) cos (n) π x d x

Take the constant out:

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

= cos (n) π \cdot \int (x + 4) x d x

Expand

(x + 4) x : x^{2} + 4 x

= cos (n) π \cdot \int x^{2} + 4 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

= cos (n) π (\int x^{2} d x + \int 4 x d x)

\int x^{2} d x = \frac{x ^{3}}{3}

\int 4 x d x = 2 x^{2}

= cos (n) π (\frac{x ^{3}}{3} + 2 x^{2})

Add a constant to the solution

= cos (n) π (\frac{x ^{3}}{3} + 2 x^{2}) + C

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What is the integral of (x+4)cos(n)pix ?
The integral of (x+4)cos(n)pix is cos(n)pi((x^3)/3+2x^2)+C