What is the integral of (2-x)sin(x+1) ?

The integral of (2-x)sin(x+1) is -2cos(x+1)+xcos(x+1)-sin(x+1)+C

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integral of (2-x)sin(x+1)

\int (2 - x) sin (x + 1) d x

- 2 cos (x + 1) + x cos (x + 1) - sin (x + 1) + C

Solution steps

\int (2 - x) sin (x + 1) d x

Expand

(2 - x) sin (x + 1) : 2 sin (x + 1) - x sin (x + 1)

= \int 2 sin (x + 1) - x sin (x + 1) d x

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 sin (x + 1) d x - \int x sin (x + 1) d x

\int 2 sin (x + 1) d x = - 2 cos (x + 1)

\int x sin (x + 1) d x = - x cos (x + 1) + sin (x + 1)

= - 2 cos (x + 1) - (- x cos (x + 1) + sin (x + 1))

Simplify

= - 2 cos (x + 1) + x cos (x + 1) - sin (x + 1)

Add a constant to the solution

= - 2 cos (x + 1) + x cos (x + 1) - sin (x + 1) + C

What is the integral of (2-x)sin(x+1) ?
The integral of (2-x)sin(x+1) is -2cos(x+1)+xcos(x+1)-sin(x+1)+C