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

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

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integral of 2sin(a-x/2)

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

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

Solution steps

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

Rewrite using trig identities

= 2 \cdot \int - cos (a) sin (\frac{x}{2}) + cos (\frac{x}{2}) sin (a) 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

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

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

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

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

Simplify

2 (- (- 2 cos (a) cos (\frac{x}{2})) + sin (a) \cdot 2 sin (\frac{x}{2})) : 4 cos (a - \frac{x}{2})

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

Add a constant to the solution

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

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