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

The integral of (x+2)sin(x/2) is -2xcos(x/2)+4sin(x/2)-4cos(x/2)+C

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integral of (x+2)sin(x/2)

\int (x + 2) sin (\frac{x}{2}) d x

- 2 x cos (\frac{x}{2}) + 4 sin (\frac{x}{2}) - 4 cos (\frac{x}{2}) + C

Solution steps

\int (x + 2) sin (\frac{x}{2}) d x

Expand

(x + 2) sin (\frac{x}{2}) : x sin (\frac{x}{2}) + 2 sin (\frac{x}{2})

= \int x sin (\frac{x}{2}) + 2 sin (\frac{x}{2}) 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 x sin (\frac{x}{2}) d x + \int 2 sin (\frac{x}{2}) d x

\int x sin (\frac{x}{2}) d x = - 2 x cos (\frac{x}{2}) + 4 sin (\frac{x}{2})

\int 2 sin (\frac{x}{2}) d x = - 4 cos (\frac{x}{2})

= - 2 x cos (\frac{x}{2}) + 4 sin (\frac{x}{2}) - 4 cos (\frac{x}{2})

Add a constant to the solution

= - 2 x cos (\frac{x}{2}) + 4 sin (\frac{x}{2}) - 4 cos (\frac{x}{2}) + C

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What is the integral of (x+2)sin(x/2) ?
The integral of (x+2)sin(x/2) is -2xcos(x/2)+4sin(x/2)-4cos(x/2)+C