Question

Integrate please show work

x dx 2eº* sinv 5 x – 3e2*sinV5 x 15

Answer 1

First we derive a formula of Integral I: Je „ax simbx dx Integration By parts da to' = Simbe le atax - Ja (simbx) (Jeax dx) de -focosbe I- Simbi e ax e ax ax + + c' I - Simbi e ax - sear cosbx dx + c -b I= Simbi e an a MC cosbx fear de focosby e at du +C I = Sinbu eax cosbx eax Pio ea fesinbx)X6 bean de] tc' a I. Sinbx ear -b Х cosbx eak - bx b J Simbre e al dx + c a a I- Simbi e al cosbx e91 -6² I + c It 62 a2 I = eque sin but cosbx ean + c' e as Ilit az asin be - bcosbuc) +c? b coston) a2 I 194762 ear Sinbre- b coobn (as a 2 td a2

I = eax a simbn-b Cosbn (ar scoba) * et le a²162 cl ܐܘ Is eax (asumbu-b Cosbu) + G a²+62 where c'az atb? = Ců Hence Constant Jeak simbeche a eqr ca singx-b cosbx) +C a²+62 Ques T- =J-dess sim 15 oC - Bedt sin 15x de 15 & ex sin 15x di I = H 3 eax Sm 5x dx I= -2 r5 jes e 3x sin 5x dx 3 15 Jeax 2x sin 15x dx (using above formula) egoe (3 Sın v5x-15 COS 152 )-3 eQx (a sm v5x - 156952) a ((3784685) 99 15 (846530) +C

I = e34 (3 Sin 15 x - √5 cos (5x) ear (2 Sin 15 5C- &5 cos 15 x) + c V5 (4+5) 3 V5 (9+5) I e 3x 3 Sin 15x -55 cos 55 x) 3 por Cosm15x - 15 los 5x) +C -8 1455 985 +6 I.-1 -I e3x ( 3 sin 15x-15 cos 15x) - I ear ca sin 15 x - √5 cos15a 915 315