Question

Evaluate the following integrals. If the integral is divergent, state so. Make sure to clearly show all steps for full credit. (a) | 22 In zde (b) ſ sinº cosº o de (e) [ 74 - gyva dy (4 - y2)3/2 t •dt

Answer 1

consider I = x²lna da By using , By Paots, we have smaffix***] - 50-80 (12) enx. 3 ارى ا 3 D x 3 loa - fx de 3 1 Pena tc where b a²enada = ₃. Bena 5x + e is integration constant. S sino caso do in. Is sino ci-sino.cosodo consider caso.sosodo - part put sino=t cosodo=dt I- St. (1-+²).dt 8 11 +7 + H tc

a I = +7 to ㅋ Resubstituting t=sino sinto 7 ie. sino.cos²o do sino to g I sinto I sinto- -singo c) consider dy I = (4- y²) 3/2 put - 4 ² = 4 sinto y=2 sino groso do 1 - s 2cosodo (4- 4s in 20)/ 2 coso.do 8 (1-sino) 3/2 coso.do 1 do coso & S seco do

! I c 4 tano Resubstituting, if y=2 sino o = sint (4/2). I= tan (sin (1/2)) t.C that tan (sinta) ol 4 4/2 Jin (9/2)² tc 4 - 42 4 I - 1 le. 1 4 y 54-42 (4-43312 1 (4-42) 312 4 to - oc 1 (4-42) 312 43

(Р tet a I = dt it te OL I dt 2 etti 3 [lone 14 22)] la ci+al) -en (i+a%)] 2 = { [0] I a 7P it t² 7 +P lim (o) HH² 00 ire. dt itt²