Question

∫ sin(5x/2) /sin(x/2)
is equal to:
(where c is a constant of integration).
(A) x + 2sin x + 2sin2x + c (B) 2x + sinx + 2sin2x + c
(C) x +2sinx + sin2x + c (D) 2x + sinx + sin2x + c

Answer 1

∫ sin(5x/2) /sin(x/2) is equal to (C) x +2sinx + sin2x + c

Answer 2

∫sin(5x/2)/sin(x/2) is equal to (c)x+2sinx+sin2x+c

1.Let I=∫sin(5x/2)/sin(x/2)

=∫(2sin5x/2cosx/2)/2sinx/2cosx/2

[multiplying by 2cosx/2 in numerator and denominator]

2.

=∫(sin3x+sin2x)/sinxdx

=[therefore 2sinAcosB=sin(A+B)+sin(A-B) and sin2A=2sinAcosA]

3.=∫(3sinx-4sin³x)+2sinxcosx/sinxdx

=∫(3-4sin²x+2cosx)dx

=∫[(3-2(1-cos2x)+2cosx]dx

=integration of [1+2cos2x+2cosx]dx

=x+2sinx+sin2x+C(integration of 1 is x

integration of 2cos2x is 2sinx

integration of 2cosx is sin2x)

therefore after applying all the integrations we get = x+ 2sinx+sin2x + c