What is the integral of-xsin(xy)-2y ?

The integral of-xsin(xy)-2y is -y^2+cos(xy)+C

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integral of-xsin(xy)-2y

\int - x sin (x y) - 2 y d y

- y^{2} + cos (x y) + C

Solution steps

\int - x sin (x y) - 2 y d y

Apply the Sum Rule:

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

= - \int 2 y d y - \int sin (x y) x d y

\int 2 y d y = y^{2}

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

= - y^{2} - (- cos (x y))

Simplify

= - y^{2} + cos (x y)

Add a constant to the solution

= - y^{2} + cos (x y) + C

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What is the integral of-xsin(xy)-2y ?
The integral of-xsin(xy)-2y is -y^2+cos(xy)+C