What is the integral of xcos(x+2) ?

The integral of xcos(x+2) is xsin(x+2)+cos(x+2)+C

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integral of xcos(x+2)

\int x cos (x + 2) d x

x sin (x + 2) + cos (x + 2) + C

Solution steps

\int x cos (x + 2) d x

Apply u-substitution

: \int (u - 2) cos (u) d u

= \int (u - 2) cos (u) d u

Apply Integration By Parts

: sin (u) (u - 2) - \int sin (u) d u

= sin (u) (u - 2) - \int sin (u) d u

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

= sin (u) (u - 2) - (- cos (u))

Substitute back

u = x + 2

= sin (x + 2) (x + 2 - 2) - (- cos (x + 2))

Simplify

= x sin (x + 2) + cos (x + 2)

Add a constant to the solution

= x sin (x + 2) + cos (x + 2) + C

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What is the integral of xcos(x+2) ?
The integral of xcos(x+2) is xsin(x+2)+cos(x+2)+C