What is the integral of xcos(x)-xsin(x) ?

The integral of xcos(x)-xsin(x) is xsin(x)+cos(x)+xcos(x)-sin(x)+C

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integral of xcos(x)-xsin(x)

\int x cos (x) - x sin (x) d x

x sin (x) + cos (x) + x cos (x) - sin (x) + C

Solution steps

\int x cos (x) - x sin (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) d x - \int x sin (x) d x

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

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

= x sin (x) + cos (x) - (- x cos (x) + sin (x))

Simplify

= x sin (x) + cos (x) + x cos (x) - sin (x)

Add a constant to the solution

= x sin (x) + cos (x) + x cos (x) - sin (x) + C

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