integrate e^cosx(xsin³x + cosx)/sin²x w.r.t x

Hi Suchita,

Split this integral as ∫e^cosx(xsinx + cotx cosecx) dx

= {∫x e^cosxsinx dx} + {∫e^cosx cotx cosecx dx} ----------- then integrate both by parts seperately.

= {x(-e^cosx) - ∫(-e^cosx) dx} + {e^cosx(-cosecx) - ∫(-cosecx) e^cosx(-sinx) dx}

= -xe^cosx + ∫(e^cosx) dx - e^cosxcosecx - ∫(e^cosx) dx ---------- (Now ∫e^cosx dx cancels with -∫e^cosx dx)

= -xe^cosx - e^cosxcosecx + constant.

And that solves the problem.

Hope that helps.

All the best.

Regards,

Ashwin (IIT Madras).