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

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

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

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

- x sin (x + \frac{π}{3}) - cos (x + \frac{π}{3}) + x sin (x) + cos (x) + C

Solution steps

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

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

\int x cos (x) d x = x sin (x) + cos (x)

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

Simplify

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

Add a constant to the solution

= - x sin (x + \frac{π}{3}) - cos (x + \frac{π}{3}) + x sin (x) + cos (x) + C

\int \frac{1}{y ( 1 - \frac{y}{m} )} d y

\int 5 t^{4} d t

\int \frac{x ^{2} - 12 x - 1}{( x - 1 ) ^{2} ( x ^{2} + 1 )} d x

\frac{d}{d x} (m x + c)

\frac{d}{d x} (e^{x} sin (8 x))

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