(i) If a differential equation is ,

then y(I.F) = ∫Q.(I.F)dx + c, where I.F = e^∫Pdx

given:-

This is a linear differential equation, comparing it with

P = cotx, Q = x² cotx + 2x

I.F = e^∫Pdx

= e^∫cotxdx

= e^log|sinx|

= sinx

Solution of the equation is given by

y(I.F) = ∫Q.(I.F)dx + c

⇒ ysinx = ∫(x² cosx + 2xsinx)dx + c

⇒ ysinx = ∫(x² cosxdx + ∫2xsinxdx + c

⇒ ysinx = x²sinx + c