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

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

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integral of xcos(x+y)

\int x cos (x + y) d x

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

Solution steps

\int x cos (x + y) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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