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integrate ecosx(xsin3x + cosx)/sin2x w.r.t x
Hi Suchita,
Split this integral as ∫ecosx(xsinx + cotx cosecx) dx
= {∫x ecosxsinx dx} + {∫ecosx cotx cosecx dx} ----------- then integrate both by parts seperately.
= {x(-ecosx) - ∫(-ecosx) dx} + {ecosx(-cosecx) - ∫(-cosecx) ecosx(-sinx) dx}
= -xecosx + ∫(ecosx) dx - ecosxcosecx - ∫(ecosx) dx ---------- (Now ∫ecosx dx cancels with -∫ecosx dx)
= -xecosx - ecosxcosecx + constant.
And that solves the problem.
Hope that helps.
All the best.
Regards,
Ashwin (IIT Madras).
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