SECTION- A adt acos(logc) + bsmCRosc). Shes ht cytcyi amd x.ynat(2nt1) cyn t6yn O y acs(logz) + bsim(og x) d = -a sim Sogo.1 t bcos(lag). i dy - bcos lag)- qsim(log) d d d y =- bsim (Regc). 1 - acos(leg x). t xdy dx yatyity= o Pavel Ditferenkiang n mes by Lesbnite's eorem +[xy1 t n.yn]tyn = O ymtatnynt n(-)yntytitnt=0 ynat Rnxyntyntn(n-1)gntnytO cha+ Rnti)cymtt Cn(n-1) + nti]yn=0 ynat(Rnti)cth yna+at1)x ymt(tgn = O rovel (e))" log<(1+ (1tlogz)³ by -(z - -1 x)leg (1t -(14Rog sc) c=y=ze get z Jady - wRen (1+lg) -C1+logs) get e(4) ea3)im eqSubstAting C4)3a-(1+lag) C34lagz (3) foa'ue get sespect (3) Difteeniating oith (2) z)lag Cit - =-(i+ ) lsg dz alse (2) lsgc) Ci + l e gy) + (1+ 2egz t 4 ) () ()Rogy [z ()t t Ot l=O aespete qet y'we gsides e )R boRRogcoim on Diftenen (i ng Taking yc y Se. Sl SECTION- B Hence frnd Salogsin x dx Se:0e hcue I -f ay cos xda Acting eq (i)and e(2)wget 2I= simx cOSx da - lag (sin2x) do loy sin 2<d - g legR Put c=u , . 3 da du Then 2I= 1f ogsim udu - Roq K -2 log smudu- og2 Sim dx - loy?-ge I-"(r- ) log Simx) da --4) Addimg i nd 2 e gef I r(G Rog2) hen. and Yfunctien and ffnd relalion betueen Sol: Defmion. The Prstand SeLond Euleian Itegpals tich Core aso caled Beta and Gamra funcionS respe ively ae defned asfeles PCm.n)fc-de and n - "Te nda. F Cm.m) is read as Beta m,n and n is readas Gamna T Hene e quantkies mand noe potive mbeS húch may sn may net be iepals Relatom beteem Beta and Gamma funchons. we hnes hat edy = n g e dy Fmemdedc Mallpeyig beth sides ef e(4) by e we get Fn.cme-tn-4 Integettng boh sicde usitl orespect foLsitkin lits c=0 yga)-dy tox 0 , e ha a(Trtm)-(ytl)x d = Wence (1tymi Cby putingl-ty and'mtn ine) AShe help of his resulf and ea 3)ue ge fifmr(ntm). Tty)m,dy r(mtn)yn (ty)mta (Cmtn) m2O 2m-i We kmou d:m n-0 simm-cosn RrGmtm) sinRmede-(m) r ) ar (mt) T(m) 2r(mt) Agun puting ysinm-1 in e C1) egel am-t ode=(r(m) 2r2m) Ssin2e)Qde -(rCm) je. 2r2m) RAng 2e= and 2d0 =d,wege do = (rfn) 2r(2) m sim m-1 dt(rCm)) 2r(m) Sim 2m-i d 2Rn-i ( r2n) Fauafing tuo values 2r(2m) s i o defom e Rmd(3) T(m)) ar mt) Hemce f(m)r(nt) NG