Question

solve integration of sin(5x/2)/sin(x/2)​

Answer 1

Answer:

The answer is  x + 2 Sinx + Sin2x + Ic

Step-by-step explanation:

Let x = 2t

Then dx = 2dt

Sin 5x/2 ÷ Sin x/2 = Sin 5t/ Sint = Sin (4t +t)/ Sint

= Sin 4t . Cost (t) + Cos 4t . Sin t/ Sin t = Sin 4t .Cos t/ Sin t + Cos 4t

= 2 Sin 2t . Cos 2t . Cos (t)/ Sin t + Cos 4t = 4 Sin t . Cos t . Cos 2t . Cos t/Sin t + Cos 4t

= 4 Cos^2t . Cos 2t + Cos 4t = 4 . (1+ Cos 2t/2). Cos 2t + Cos 4t

= 2 Cos 2t + 2 Cos^2.2t + Cos 4t = 2Cos 2t + (1+ Cos 4t) + Cos 4t

= 1 + 2Cos 2t + 2 Cos 4t

I = ∫ Sin 5x/2  ÷ Sin x/2.dx =  ∫( 1 + 2Cos 2t + 2Cos 4t).2 .dt

= 2 . [t + Sin 2t + Sin 4t /2 ] + Ic = 2t + 2Sin2t + Sin4t + Ic

= x + 2 Sinx + Sin2x + Ic