Question

. 1. (Indefinite Integrals; 33pts). (i) Find the integral 42² 3 (x +9)(x - 2)(1-9) fin) Use a suitable change of variables to find the integral tan(I) da. (in Use integration by parts to find the integral

Answer 1

Given integreer is 144. 3 9nt du (+9) (-2)(n-9) By partial freet on de composition. Let un²-3 (449) (1-2)(n-9) un A + B + C D - (nota (0-2) cuta) 40-3= A(-2) (169) + B1.449)(n-a) + C(79) (-2) like Equating the co-e foi cient of teum ATB t.C = 4 - 2 7A-110ZO -18A-81 B 418C = 3 € multiplying egn. O by 18 and add in ogn. W den 184 + 18 R +180 = 72 - 18A8/B +18CE-3 -63B+366 = 69

Multiplying Subtract can ③ by 7 then from egn- o 0 - 0 7A-11C tot TB + T = 28 aon - B - 180 =-28 & TB + 180 = 286 multiplying con (6 by 9 and 1- add in lean 5 63B 63 +1620 252 + 360 = 69 & 2 1980 = 325 321 - 1070 798 66 put (= 107 neque weget B + 188107 - 28 66 BZ-13 7

puti B=-13 and re=rot In ogn 0 we get A10 U2 low The negal bе а не suzdu I (ntain-2 (na) = 11oT, -13 No6(nea) 20 W -2 O +107 1 42 10-9 du Elo7 In (249) — 1311(72) 4107 10cm9 Theree Gove t us 66 77 पर याबा 70-19) (4-2) (-9) 9. = 66 107 Incinta!) – 73 \n(112) t 207 1n (12-91) te 77

E J font = J Simu an o substitut u=con - a duz - sinn du cosu Le due -t du - ! du z - Inca) undos uostitution ut con - – Inclosu) Therefor tanne - In (1 cosul) f = nilseente ( 1nt Integrate by - Integrate by parte f = 1n2cm) gal fa & they g = X n2m) - 72 Inow on

= n lntow -25 lnx du - n1n? (1) + 2 [ a Inn -x] I landu 협체거리에 Тkoefox du= nln n anatan