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