Question

W60. Compute (sin x + cos x)(4 – 2 sin 2x – sin? 2x)e" dx sin 2x where = 6 (0,7) Mihály Be

Answer 1

we will integrate Numeratos ad denominatos separatly. By re-arranging Murrelatos Sacsinsin) + casus) (-2 sin 12M) - Sin (2 + y de. Apply linearity Sea oso lo trocul) (2 min ( 20 + stor?]m-4) ou now sohning MO + casse I (25inza + sin 2x -41 d. reweite S(Re Siriu) sin(20) + z elastussis (24) + Sim (2) ne" sinu – he'sinin + sin²12) nelaon ayeu cas aldu - 2 Setsin(u) sims (24) dut a se casin) sin 12W dn-ufe sincu te could + sin(2) Shesis and dut sin (2) & nel cas y de Now solving Seusinky sins (24) au. Apply product-to-sum falmulas sin(u) . Sinly! I las lynn) - Carly told, sint (M) = 1 (1 - Cast 24 h. Cars (u) looly) = & Caslyta) & cooly w), Cas? (n) = I cars (21) + 1), sin end Cosly) = (sin (ytul in 19.4), lash) sin(x)=1 fin (24) a fel cao 3x Loon du empand

Stran a las 134) an Apply Cimenty = 1/2 Setras mad dhe I See Cooland dy. now sowing Coolut du. we will integrate by parts twice in a row Se gl= fg- Stig first step ta loon) t'a-sinn, - e coola) -s- ensin (nd du second time. t = -sinca), gl=en d'= - Casa e "cas - (-e since) - Such cas cul du) Apply Grearity. are et caslu) - fen sin u + sex casiuldu. The integea secas (ul de appears again on the right sid of the equation, we can solve fal it - e" sin (M) + e cas lw.

Now solving Cas134 du we will integrate by parts twice is a row Stg afg-Stg. First time. fa Cas (34) ( ge ta -3 sin (34) - e cas (34) S-30 sir (suldu second time -38in (30 gl=en Ho acos (37) e cas 13n) -(-34 sim (3n) - S-9 et cas 134) du) Apply Lineases = e coolan) - f3e" siss (34) tase Meas canlan). on the right side of the The integed se cos (34) dhe appears again equation, we can so we fal it = 3ê sin 13 ") + e cas (34) 10 I see cas (30) ad. plug in Solved Pategrals sex casculdu - e sir (n) + e cosm) 3er sin 134 tl cas(34) 20 니

en son 3 ut sinn du Now solving Selcascaisin (24) da IN The sitegeal sehinlonley appears again Applying product -to- dum falmulas. - S DEL- expand S ensin 34 + evoirslan Apply linearits & se "sim (30 cut & se sir in du Now soking First Time ta sim (31 1 8 t = 3 cas (341 a em sim 1341-f3e" cas 3n du t= 3 cos 3n t'= -9 sinn, e Yin 34- (36" cas34 S-ae sin (34) cul Apply linearby é sing - (3 "Cas zur q Sensinsway on the right are y the equation vol can sowe fal it ad se sims 1311 CM g=cy second time as a tojolate swing ka gl=en

e sin (34) - ze con 3d 10 Now sohrein se sin(n) du ) ta sinn gle g=e" t = con, esin cul secas (nidu second time to coolusi sinn, ga en esin (M) - (et casino - Se simul and Apply linearby e sin (1) (eloslul & sesin curdul. right side of the The integerd se sin man appears again equation we can solve fal it en sins (M) - el cas (4) 2 in soled integrals plug & Sex sim 130 cm + & Sefir (war egin 134 -3 e cosla + esin (nd - e "cas (M) now sochung ។

Sie sir (nd te "cas (ul de Apply inearity tas the a fel für and do the basind da. with in the sum , apply integration Sen cas cul de iste's fg - Stg. fae" g'a lasin) e مال m] a esin end - Jeu fins (4) au the sin cul de The integral je sin (M) du carches leaving only & ension in Now solving. Sue "sin end an. Integede by parts ist g' = fg - Jf g. ton, gla e' sro cul ga "Sin (12 mt cas / а rretain (m) - el cason - Senin- erson ex e now sowing e coseul en fin in 2 Apply & rearily Secas (udde, Sensorcuddu

with is the sam, apply integration by parts for Set was in) da; If gl= dg astig. fa lassul - glo en ta -sin n) ga" + - elason) + 2S-e" pirn dat Sesunn du The integeal £ s engins in de cancels, leaving ay - e" Los M] plug in in sowed integids e Castall n le sir (M) - Sesin in) - e" casin) dr. ellas and a mie" sin (m) é los in pula now solving Son el cas (Md du Integrate by pauls Stapleto - Sig tan, g'= el casal dla !, g = esime + en casa What's =ule Sirs (M) + "coolul] - set sin (m) + el conry Now sowing Sevgin and + evron Apply = I Se sisman & & S et aas in die du inearelas

with in the same apply integution tas Semeas w dua Sto's dg -Si g. 32 ta en 7 g' - coolul = sin in g I enfin in sesis in aux 1 se sus mil du integral & sesini de cancels leaving of - e ginen) berganti wa plug in Jowed integrals, are "sirs and teleaolu) Se fins a te cay CH Fanm a nie sin ind te casu plug in Sowed integras. a fengis(n) sin (23) du tz ſe "casem) Sim (21) du-y ste fin (w tetas i di + first and Sue Sin in an & sir? (2) f netaoculdu 3e" sin (30) telas (34 ensins (34) - ze coo (34) (2) Uhe sol 10 го न - en con esinn -elasy 2 t e sind fe cost tsin² (2) ve Sinin Inderasy helsin (w + sin(2) ecolul plug in sowed integends. - Sexsina & Co (Mi) 1 (2 dis (24 tsin? (214-41 du

ze sin 134) + el cas ( 3W e sin (3 9) - ze coolan 10 16 sin 70a (etsin (m) te loomit sin ini tetasin d - Sin? (2) & cesis (nt - eas in elbin (n) – en lasy tin (2) e" sin in & yelsin (n) a sim ² al élasta The problem is solved Searsin (12 t cas (ni (1-2 sim (24) - Sinh Tylda 3 en singt elas 34 e singu - 3 e con 34 sim 2x ésinn te kogu 16 'sim te casy sin za te sin n-e "last esinneasa + in? edina ty elsin (1) - sin?ele cas (m) + c Rewrite and simply. 1/2 sims 134) tucasian)- Salsin? (2) (2x - 1) -61 Sin in + tirin cosy 10 + Now denomatol. لاله 3- 37 وضع 2 (e" sind & coon e 3sin 342 +60szul प 0 RO + eysir (nl-casin I sin 262) (nel girim) + (lauercast sir 2/2) (1-1) 16" sin(n) & neu casn theysin ind trosu 2 t 2 ensinn - Cosy) to

10 Simply ad se wale e ce sirs (34) tulos (34) + (1 10 sin? (2) utstin (2) +30) se - 5sin al loss! Now denominator I sin an du. subsit kad l = 24 du 22 7 = f du. Isinulay sain ruldu preparing for helsti tule =fCl - Cos? (47) sinculdu. sin (u) a du= -1 sinal subsil - Celulasan Av² il du Apply Credits asuhdv fidu Now sowily tim se do 12 own to No co solaris sidu -ازاد 19

plug in sowed inleges sordu-sidra undo substitution & Coscul Cool31 a los3 ru} plug in solved instegids Ź s sim3 cul du 1003c4] 6 coolu indo substitution U = 24 2003c2M) 6 problem i rowed The sin3 (anddy - Cas 12W +C es 3 (20 Rewrite and simply acas Caul +G cas (64) 24 denominatos. Allumeedte 2 کیا 10 (2 din 134) +4 Cao (34) + (-10% in 2(214) + 5 sinfz) +30) sin in - 5 sin? (2) Coscullt 의 16 Cos 16u) a Cas (24)