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SRI VIDYA COLLEGE OF ENGINEERING & TECHNOLOGY COURSE MATERIAL (LECTURE NOTES) i Onir- it G | CENTEGRAL _CAL.cunug. | Tihoduchin : | Takagral calculus & the ludy nh! jondeng 0 jundiin bated on ‘tho injermnaliin about tb Ge 4 chalge. 7 | Tho indore inkegpnals | Formule : 1) Jurdx = ot Be n+ ED) jx dx = a 3) Jo% du = o%+e 4) [yg du = avec, 5) J sinndx =-cozxte 6) J cos adx = am x +e +) J aoc®x dx = tanx+¢ 8) J cosactx dx =—cobkx +e 9) J docx banxdx = Beex+e to) J cosae x cot xdu = —Cosae x+C 1) f cos ha dx = sin hate 12) J ain hedx = cos hate: isp | '__ dx s ban'x+¢ teo% dx = gin'x+e - |S 5) J =: dx = tog (x4 Vata) +e 1b) { m= Log (x+Yaxter ) +e x1 m) J wo as AX = BOO HO ya pce MA8151- ENGINEERING MATHEMATICS I Page 1 SRI VIDYA COLLEGE OF ENGINEERING & TECHNOLOGY COURSE MATERIAL (LECTURE NOTES) | Phoblome e ) Evaluate J (04-8 nea) dx. | Seth: | Sato. Ad 8g tg So fad as me \ J@-4% ++tx-2) dy - x 5 (22) L (as) ax tc = xP _ ety ae _ ante 2) Evaluate f curaycautde. © a : §€uta) Cautt) du - J(aurt+us uty) du = J Cawtsquts) du = aut +9 us +AU tC 3) Evaluate : | xSraxt] dx y* f x8taxtl HS 4 2K | | dx = \(Sr* pet oy) dy bya [&+4art) du =< [Ger ae em de " mx 3 oa slogan tax? y eo: 7 =3 \ \ = lo Co a 4) Braluate [Gur xB 1) doe gi: <2 Ie sla [ates 2 Fo) de - { (bao 4x84 E ay) dx _ 3}9 sla = Paty 2S + Bs SF logn FR+E =i sla Bla. d 3 5) = ob yD yl) 2 ote teananee mH 8 a d 5) Evoluate : [Grd yrdx. goin + orl [orgy de = [tga eld = Ba Eros x | = 2 _ jb +anee- g Ww MA8151- ENGINEERING MATHEMATICS I Page 2 SRI VIDYA COLLEGE OF ENGINEERING & TECHNOLOGY COURSE MATERIAL (LECTURE NOTES) 7 « ! du = J sin* x +0062 dy e Ur z 2 j gin®x coax @in?x cos? % r gon *x oes dx + i ~ am x cortx Sm*x cw - | Hines ) da - cas x ain? X {Goot 4 casoc® x) dx tanx- Cot x4+C- \ doe 9) ee 14 8inx | $5 He ex | LE 8% doe | ome 7 | Wwsinx = 1-8 = [ 12-89% _ du = | Se dx ;—an*x cost x = ! — 8x V doc 7 J Coa? “con® x J{goe*x _ kann -go0 x Jdx = tanx— Secx+c 1) Evaluate j tos yy a 1— tn x Be Oacsby ae ~ b= stn dy | (i-ainx) C148) 4 | | — 8th x aa J 1-8 x C1-8Hx) i JUtsm x) dx = “ cosxte D) Evaluate Jf (Tyemax dx “ho: J {| ain *x 4 con 2K 428i BX Cad _% det J (gi % + cos x) * dx | (ain + cos x) dy on - o gs! Re x [a 2 tr ig I | ~coax4+Sin H+ C. tt MA8151- ENGINEERING MATHEMATICS I Page 4 SRI VIDYA COLLEGE OF ENGINEERING & TECHNOLOGY COURSE MATERIAL (LECTURE NOTES) IQ Fvalaal J Cana acot a) * dx ) yi: J C tons ~acot x)* da = { Gontx 4 Cot? x 4 Lane cot 2)dx = § (see! +4 Coonoe % — 1) 4 tana ty & | Cece x +4 cosoc 2% -9 ) dx | | | = tana — scot x-4x4¢. 1) Eval wate ic + 0% =) da | ile OED du = 36in'x 40% sande | yen™ 1) | (corba + | + —,) dx | Papuan bane \ 4 | G I ral 4 - § {oor he eet Ve dx = Sim ha + 800! +ton'x+e | Propontias 4 dept infogna | pet j and i, cae continuous ponclion DJ pede = Py corde a a v | & yoo dx = J jrooda * | b b b joodxr | qooda = Jyoodxsazeck © o 9 ee ee | 7 Z 14) J Joodx = J Ca-x dx a Qa (5) G) Ty 400 a an oven punclisn 4 ot, then [poode= a [yond 07} i & an odd junelién 9 x, then i poodn = 0 6) J yoods = Jf Carb-m) de . a MA8151- ENGINEERING MATHEMATICS I Page 5 SRI VIDYA COLLEGE OF ENGINEERING & TECHNOLOGY COURSE MATERIAL (LECTURE NOTES) My @ Lod (aes ag bine die 4 OSH lay e Sin x 4 COs % Con x 1 Bin» | ws “ila | Sth x4 Cos du = | ada “| 8m x +Con% 6 ° oe = ta" : | Te | ae A Ts 5) Evaluate J 008% dx Cord l* ses dx Bch % +008 X J tt tans ° st Wa J of Te 002% de . | Sth % 4 Coz % —m) 7 oe T= | cos (E-x) de x am (E_ a) cos (E -x) We = | eS doe os ie 8th x +CosK ® Tie . | C4 (2) =) & = J osx + 8inx dx Sth H+ Cos XK 5 Fae > ar = = ats Maye - J x = x > an 7 6) Evaluate a dx Cor) ae ! dx. hx + Ycos SN Gan oi | aan ere hot T= | —¥ceex__ des ay 3 Yainx+Vooex | ie | dx 3 (2 | J Neos +8 ) 1) teow ieee Crea moray ars { — Lee da + J gine dx | Yainx + eos x Yoosx +V 8h x | a tO Yeon + Van we — | = | See Sf de = Pret | 2 \snx + Ycosx A a 4 MA8151- ENGINEERING MATHEMATICS I Page 7 SRI VIDYA COLLEGE OF ENGINEERING & TECHNOLOGY COURSE MATERIAL (LECTURE NOTES) 4) Evaluate ws Tee Got x aly pres | | 5 Veo x 4 Vtanx ° Sein. Ws = wot t- | Wee J Yeat x +N tar x so) - 1" [et E-*) | | ee ee ey We - = { er dk ow ¢ Vtonx +Vcorx whe = ars Yeotx + Vtanx nie TI | as EE de = J dx = [a | ¢ tan x + Ceot ae ‘ 2 @at= Tayi . 8 SL - 8) Evaleal f Vac dx » Vec+Ya-x hot 3 - J Voc du. sc _ Ya + a-% = i Va=% dx % Na-% +a-ta-x> Te f am dx — la) 5 Na-% +N« oO Tw d e a4 dx Cr+Ca) => oT = | fi x4 f ar LR a 7 ° _ Vor + To-x dy sf dx = [xJ sea 2 Vor + Vax ° Ts @ | Ts “ %) Evaluate J log (tanxd dx . | ais" ay, | dot T= f tog Ctanxddx 9 4) Ls Pte {ton (Ex) | dx MA8151- ENGINEERING MATHEMATICS I Page 8 SRI VIDYA COLLEGE OF ENGINEERING & TECHNOLOGY COURSE MATERIAL (LECTURE NOTES) " We 1 Evaluak J cos % dx @ 7 ~The Galo : ails dct T = f cor xdy Yy Key f 0x) = Coa x 1” = Cos (-%) = Cos x pe pox) & on oven junction “He Wa _ f ceosx da = 2 i coaxdx = 2& [ee z ~ Cos °| Ts ° = a(o-i)= 2 13) Evaluate r 3 as dx J jrat-37z65 o S6in § dct T = j fx) dat 3 103 fath-2d4 4X? a by Catrb~%) AG 7 J j fatb Core ad4 4 Catb—) a3 W wes (atb~ 2) — dx —» ¢4) 7 J joo" jcorb-%) a U t ‘iy 5) c 5 CNC) “S ote J [OO +4 Cotb-9) dy | a j cart xD + AOD i )s . " \ ba J du = Yo), = Da o Te b-a a 7 i) Fraluots { gine x cos x du i ist T= | Bintx coax cx ° yoo = Sin2« cos ?x {CT = Bin OT a> Cone TF =) | “ 2 3 —— = &m*x (- cosa)? =— gunix con? = —f Cad MA8151- ENGINEERING MATHEMATICS I Page 10 SRI VIDYA COLLEGE OF ENGINEERING & TECHNOLOGY COURSE MATERIAL (LECTURE NOTES) ° { Le " - aa (0) e- | 8m 8x costa de = 0 i J 4 odx =o “4 Caf Row 2a =a T 9) Evaluate J x log C8tn x do . we: hot T= fx log Cam x) dx. 9 «1 (ar-9¢) bog [Sin (1-27) dx . : fyeods = Pyeooe 7 Te J (T-%) se (8marddx .—s (a) TT Ss Macao ar- [x log (std dx + J Cr~%) log C8thandx 6 ° " [ Gann) lea Cad x da. 7 We Jor leg Comardx = at J Ing Camm xd dx. ° q = [yee ° it a a Pyeoda 4 (20-20 434 Ws =7 J leg Csr 0) dx ve log 2] 7 *y tog (sirxydx =~ leg of z 2 <1 i + tog (25 = a log Substitulion Rule Titooa ation by dubstilulion Con) change 4 tho idepondont uni, Typo Abe JEyoo]" pG0 de Cond $$ hood {Gods . Substilute u ~ =}00 ond ‘thon proceod . 4 pe|— ond 4 Phoblarns. 1) Sealuote 5 Cax+b)” de . S60. MA8151- ENGINEERING MATHEMATICS I Page 11 SRI VIDYA COLLEGE OF ENGINEERING & TECHNOLOGY COURSE MATERIAL (LECTURE NOTES) | ra} Pub ust : ane Vy min 6 e pe Pe ee a > ae = adu | i i | uiesadu = 9 Juda = a (=) ¢ 3+) 4l = i(2- ae 6 wae 2 3 Cuye) Bt. Alg A 2 6) Evaluate | x2 da | oe - ais: oy hot T = i ry roms | Pul u- fers =Sdued \ -s edu .t dex “ * ators bat r=] utande 2 xeut 5 o> ute tuts)? ets ut a5 -iou® Te J (t+ a5—iout) acdix = ue a6 Tee. = a [4Fs acu ° | vie ¢ asa Vs = 3 [Gos yg DES 4 as Cars) 4 ce . a 3 a (x* -. 3 (84 an) dx +) Evaluate Jus (+t +1) Soin: dot Te Jie Cudgantyiy® Cee and do - Put ws xrearxtti 7 du=GxF tere 4 Ca Stand dx - =| du. 6434 axddy _ ad 2 _ es : W's fia w a 2 9 Jw du = 3 (2) +e = uste = CrPtaxteryea ¢ \ 8) Evaluate § ae dx | gah: tok Ee | pee Oe lot I = J fexcaye Pur U=ax-4 3 du=3dx ah = dx - —3)a ls yale pala aC | ¢ Te Jatu dee 5 Jade = Sl Aa MA8151- ENGINEERING MATHEMATICS I Page 13 SRI VIDYA COLLEGE OF ENGINEERING & TECHNOLOGY COURSE MATERIAL (LECTURE NOTES) fl | oy 42 de t =~ a | i A 3 3x 4 Evaluate a J “Spe de Bahia ee ct “ otto. Lot Ts | {eg x ca Put we as = Juda = tere es Mega)" ¢ Bog x i du= tidy. 10) Evaluate ! laid oe % Log x | hot | * Ieee we pe Pie tomlin dus LL dy | A Je du = Plog u] + ¢ = log Clog xd 40. 'D Evalual . dx | ei f sy ms 27f oie X>!1 =suso Xe ss us) 12) Evalucite Boc* Clogx)d - J EE da a Jot T J aoc ee du dot we hog x Sdu- tb du. mw J soctu du = tanut+e = tan Clag 2) Ce. Exponanlial junctions 3) Fvoluate Sora dx Bein : ket Fe J ot x® dx MA8151- ENGINEERING MATHEMATICS I Page 14 SRI VIDYA COLLEGE OF ENGINEERING & TECHNOLOGY COURSE MATERIAL (LECTURE NOTES) ¢ alogtlegas oe, : \ 6 ¢ 4 dae cre | +c 7 leq Cog a) = Llegad® 7 . tq Cog a> Jrigonoraelnc jundions 9) Fvolaale | cox? ain o do Safi hot T- Jocoz20 ane de Pul u=cose 5 du =~&ing de T= J ut tedu) - futdu =~ ur ye .— Coors) 4, ) Evaluak Bac® (Myx) * “ : J att da lek Ts Soc ® (Vx) cea Put =e + due ~ 4 dx =s-du- ou T= f see*u. t-du) = — J 8ec%udu = ~tanute = ~ ton (K)+e. %) Evoluate |e : in? x gi{n : hot I Cozx Y,, = re doe Pul us finn 5 du=corx dx _ Wk : 6 —2t Te faa du = futdu = ae +¢ =- +e = =. he 8th % 2) Fvoluok | 22 dy mls 7" dot T= Sin NX df, | = du . \ Pul u=Vx + du= _ dx => adu- -i dx ay Vx xn pee J Shu Cadu) = 2 famudu = a (-cosud +¢ - =- 2 con VK HC MA8151- ENGINEERING MATHEMATICS I Page 16 SRI VIDYA COLLEGE OF ENGINEERING & TECHNOLOGY COURSE MATERIAL (LECTURE NOTES) $3) Evaluate | aoe 8x dy 8 e 5 +4 tanx kink SP = (SS dx 5+4 tan x Pul us sts tanx ; du= 4 sec®x dx =s du = Backad: | Te ft Met fodus t [log a} +e = dog (Stu tana) +c. &) Evaluate J cos C¥e+3) de Lot T= f cos C¥e+3) de du~yde => du =de. (8thud +e Pul u= 443 5 Ts feoru. = tL Jooru-du = + = 1 ath (xst+ad4e. | 7 a5) Evaluote | @oc® Cot +1) 5 dt. | hot T= faoo*t st +) & dk Put u=bE+1 ; duaodk - Te faocku-du = tanute = tan CoEHD+C . “ " 2b) Fraluate J Citeth xd cos x doe . si: hot T= f Citain xy” cos x dx Put ws 14+8inx 5 dus cosxdu. Te j ut duc UF +0 SEL as " &) Evaluate J 800 %du - lot T= J 80cexdx = { Sacx Caoe x +tan 2) da SOC wba x . hot w= Soexttana ; dus een tense +800%x) dex = Boon Clanx +seexd dx MA8151- ENGINEERING MATHEMATICS I Page 17 SRI VIDYA COLLEGE OF ENGINEERING & TECHNOLOGY COURSE MATERIAL (LECTURE NOTES) 0] @ Probleme : D Evaluate Joe ain doe gan: hot us x dws sto % dy dus ca V = Cos Judy = uv-Svdu ~ be an Te*-o7J +¢ ws Pl, a at = ye 5 (co +e. | 2-8 ¢% foe +p+e¢ 4) Evaluate J Geos hb dt Goin ;: hot ust dv = cos ht dt du=dt V= Sthhb. b fudveT = Tuv a -Jvdu a i i kcoght de = |e sa hE], - patil dt = anhi — J) cos "wa - Sa hl ~ Leos hl — cos 0] a —I = and'- [ae -1] glee o-d'¥a}~ 4 [a8 2 {- rr Med c MA8151- ENGINEERING MATHEMATICS I Page 20 
