What is the integral of-xsin(x) ?

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

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integral of-xsin(x)

\int - x sin (x) d x

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

Solution steps

\int - x sin (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int x sin (x) d x

Apply Integration By Parts

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

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

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

Simplify

= x cos (x) - sin (x)

Add a constant to the solution

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