Question

`int (sin  (5x)/2)/(sin  x/2)dx` is equal to ______. (where C is a constant of integration).

Options

• x + 2sin x + 2sin2x + C

• 2x + sin x + 2sin2x + C

• x + 2sinx + sin2x + C

• 2x + sinx + sin2x + C

Answer 1

`int (sin  (5x)/2)/(sin  x/2)dx` is equal to x + 2sinx + sin2x + C. (where C is a constant of integration).

Explanation:

`int (sin  (5x)/2)/(sin  x/2)dx`

`int (2sin  (5x)/2 cos  x/2)/(2sin  x/2 cos  x/2)dx`

= `int (sin3x + sin2x)/sinx dx`

Using 2sinAcosB = sin(A + B) + sin(A – B)

= `int (3sinx - 4sin^3x + 2sinxcosx)/sinx dx`

= `int(3 - 4sin^2x + 2cosx)dx`

= `(3 - 4((1 - cos2x)/2) + 2cosx)dx`

= 3x + 2sinx – 2x + sin2x + C

= x + 2sinx + sin2x + C