(i) If a differential equation is ,

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

(ii) ∫tanxdx = log|secx| + c

(iv) ∫cosxdx = sinx + c

given:-

This is a linear differential equation, comparing it with

P = tanx, Q = x²cos²x

I.F = e^∫Pdx

= e^∫tanxdx

= e^log|secx|

= secx

Solution of the equation is given by

y(I.F) = ∫Q.(I.F)dx + c

⇒ ysecx = ∫(x² cos²x(secx)dx + c

⇒ ysinx = ∫(x² cosxdx + c

⇒ ysecx = x²∫ cosxdx–∫(2x cosxdx)dx + c

using integrating by parts

y(secx) = x²sinx–2∫x² sinxdx + c

⇒ y(secx) = x²sinx–2(x∫ sinxdx–∫ sinxdx)dx + c

⇒ y(secx) = x²sinx + 2xcosx–2sinx + c

⇒ y = x²sinxcosx–2xcos²x–2sinxcos²x–2sinxcosx + ccosx