What is the integral of (5+2w^3-w^5)cos(3w) ?

The integral of (5+2w^3-w^5)cos(3w) is

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integral of (5+2w^3-w^5)cos(3w)

\int (5 + 2 w^{3} - w^{5}) cos (3 w) d w

- \frac{1}{3} w^{5} sin (3 w) - \frac{5}{9} w^{4} cos (3 w) + \frac{38}{27} w^{3} sin (3 w) + \frac{38}{27} w^{2} cos (3 w) - \frac{76}{81} w sin (3 w) + \frac{5}{3} sin (3 w) - \frac{76}{243} cos (3 w) + C

Solution steps

\int (5 + 2 w^{3} - w^{5}) cos (3 w) d w

Expand

(5 + 2 w^{3} - w^{5}) cos (3 w) : 5 cos (3 w) + 2 w^{3} cos (3 w) - w^{5} cos (3 w)

= \int 5 cos (3 w) + 2 w^{3} cos (3 w) - w^{5} cos (3 w) d w

Apply the Sum Rule:

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

= \int 5 cos (3 w) d w + \int 2 w^{3} cos (3 w) d w - \int w^{5} cos (3 w) d w

\int 5 cos (3 w) d w = \frac{5}{3} sin (3 w)

\int 2 w^{3} cos (3 w) d w = 2 (\frac{1}{3} w^{3} sin (3 w) + \frac{1}{3} w^{2} cos (3 w) - \frac{2}{9} w sin (3 w) - \frac{2}{27} cos (3 w))

\int w^{5} cos (3 w) d w = \frac{1}{3} w^{5} sin (3 w) - \frac{5}{3} (- \frac{1}{3} w^{4} cos (3 w) + \frac{4}{81} (9 w^{3} sin (3 w) + 9 w^{2} cos (3 w) - 6 w sin (3 w) - 2 cos (3 w)))

= \frac{5}{3} sin (3 w) + 2 (\frac{1}{3} w^{3} sin (3 w) + \frac{1}{3} w^{2} cos (3 w) - \frac{2}{9} w sin (3 w) - \frac{2}{27} cos (3 w)) - (\frac{1}{3} w^{5} sin (3 w) - \frac{5}{3} (- \frac{1}{3} w^{4} cos (3 w) + \frac{4}{81} (9 w^{3} sin (3 w) + 9 w^{2} cos (3 w) - 6 w sin (3 w) - 2 cos (3 w))))

Simplify

\frac{5}{3} sin (3 w) + 2 (\frac{1}{3} w^{3} sin (3 w) + \frac{1}{3} w^{2} cos (3 w) - \frac{2}{9} w sin (3 w) - \frac{2}{27} cos (3 w)) - (\frac{1}{3} w^{5} sin (3 w) - \frac{5}{3} (- \frac{1}{3} w^{4} cos (3 w) + \frac{4}{81} (9 w^{3} sin (3 w) + 9 w^{2} cos (3 w) - 6 w sin (3 w) - 2 cos (3 w)))) : - \frac{1}{3} w^{5} sin (3 w) - \frac{5}{9} w^{4} cos (3 w) + \frac{38}{27} w^{3} sin (3 w) + \frac{38}{27} w^{2} cos (3 w) - \frac{76}{81} w sin (3 w) + \frac{5}{3} sin (3 w) - \frac{76}{243} cos (3 w)

= - \frac{1}{3} w^{5} sin (3 w) - \frac{5}{9} w^{4} cos (3 w) + \frac{38}{27} w^{3} sin (3 w) + \frac{38}{27} w^{2} cos (3 w) - \frac{76}{81} w sin (3 w) + \frac{5}{3} sin (3 w) - \frac{76}{243} cos (3 w)

Add a constant to the solution

= - \frac{1}{3} w^{5} sin (3 w) - \frac{5}{9} w^{4} cos (3 w) + \frac{38}{27} w^{3} sin (3 w) + \frac{38}{27} w^{2} cos (3 w) - \frac{76}{81} w sin (3 w) + \frac{5}{3} sin (3 w) - \frac{76}{243} cos (3 w) + C

x = y^{2} - y, x = y - y^{2}

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f (x) = x + sin (x)

What is the integral of (5+2w^3-w^5)cos(3w) ?
The integral of (5+2w^3-w^5)cos(3w) is