What is the integral of 2xsin(x/2) ?

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

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integral of 2xsin(x/2)

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

: - 2 x cos (\frac{x}{2}) - \int - 2 cos (\frac{x}{2}) d x

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

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

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

Simplify

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

Add a constant to the solution

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

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