What is the integral of x(cos(x)-sin(x)) ?

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

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integral of x(cos(x)-sin(x))

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

Simplify

x (sin (x) + cos (x)) - (- cos (x) + sin (x)) : x sin (x) + x cos (x) + cos (x) - sin (x)

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

Add a constant to the solution

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

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