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

The integral of x(-cos(x+pi/3)+cos(x)) is

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integral of x(-cos(x+pi/3)+cos(x))

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

\frac{1}{6 3} (- 33 x sin (x + \frac{π}{3}) - 9 x cos (x + \frac{π}{3}) - 3 π sin (x + \frac{π}{3}) + 3 (- π - 3) cos (x + \frac{π}{3}) + 9 sin (x + \frac{π}{3})) + \frac{π}{6} sin (x + \frac{π}{3}) + \frac{π}{2 3} cos (x + \frac{π}{3}) + C

Solution steps

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

Apply u-substitution

= \int \frac{- u cos ( u ) + 3 u sin ( u )}{2} + \frac{π}{6} cos (u) - \frac{π sin ( u )}{2 3} d u

Apply the Sum Rule:

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

= \int \frac{- u cos ( u ) + 3 u sin ( u )}{2} d u + \int \frac{π}{6} cos (u) d u - \int \frac{π sin ( u )}{2 3} d u

\int \frac{- u cos ( u ) + 3 u sin ( u )}{2} d u = \frac{1}{2} (- u sin (u) - cos (u) - 3 u cos (u) + 3 sin (u))

\int \frac{π}{6} cos (u) d u = \frac{π}{6} sin (u)

\int \frac{π sin ( u )}{2 3} d u = - \frac{π}{2 3} cos (u)

= \frac{1}{2} (- u sin (u) - cos (u) - 3 u cos (u) + 3 sin (u)) + \frac{π}{6} sin (u) - (- \frac{π}{2 3} cos (u))

Substitute back

u = x + \frac{π}{3}

= \frac{1}{2} (- (x + \frac{π}{3}) sin (x + \frac{π}{3}) - cos (x + \frac{π}{3}) - 3 (x + \frac{π}{3}) cos (x + \frac{π}{3}) + 3 sin (x + \frac{π}{3})) + \frac{π}{6} sin (x + \frac{π}{3}) - (- \frac{π}{2 3} cos (x + \frac{π}{3}))

Simplify

\frac{1}{2} (- (x + \frac{π}{3}) sin (x + \frac{π}{3}) - cos (x + \frac{π}{3}) - 3 (x + \frac{π}{3}) cos (x + \frac{π}{3}) + 3 sin (x + \frac{π}{3})) + \frac{π}{6} sin (x + \frac{π}{3}) - (- \frac{π}{2 3} cos (x + \frac{π}{3})) : \frac{1}{6 3} (- 33 x sin (x + \frac{π}{3}) - 9 x cos (x + \frac{π}{3}) - 3 π sin (x + \frac{π}{3}) + 3 (- π - 3) cos (x + \frac{π}{3}) + 9 sin (x + \frac{π}{3})) + \frac{π}{6} sin (x + \frac{π}{3}) + \frac{π}{2 3} cos (x + \frac{π}{3})

= \frac{1}{6 3} (- 33 x sin (x + \frac{π}{3}) - 9 x cos (x + \frac{π}{3}) - 3 π sin (x + \frac{π}{3}) + 3 (- π - 3) cos (x + \frac{π}{3}) + 9 sin (x + \frac{π}{3})) + \frac{π}{6} sin (x + \frac{π}{3}) + \frac{π}{2 3} cos (x + \frac{π}{3})

Add a constant to the solution

= \frac{1}{6 3} (- 33 x sin (x + \frac{π}{3}) - 9 x cos (x + \frac{π}{3}) - 3 π sin (x + \frac{π}{3}) + 3 (- π - 3) cos (x + \frac{π}{3}) + 9 sin (x + \frac{π}{3})) + \frac{π}{6} sin (x + \frac{π}{3}) + \frac{π}{2 3} cos (x + \frac{π}{3}) + C

What is the integral of x(-cos(x+pi/3)+cos(x)) ?
The integral of x(-cos(x+pi/3)+cos(x)) is