What is the integral of (x+y)sin(x+y) ?

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

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integral of (x+y)sin(x+y)

\int (x + y) sin (x + y) d y

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

Solution steps

\int (x + y) sin (x + y) d y

Expand

(x + y) sin (x + y) : x sin (x + y) + y sin (x + y)

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

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

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

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

Add a constant to the solution

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

What is the integral of (x+y)sin(x+y) ?
The integral of (x+y)sin(x+y) is -xcos(x+y)-ycos(x+y)+sin(x+y)+C