keep getting stuck

Question

for integrate e^(bx) cos x

i get

= cos x e^bx/b - int -sin x e^bx/b
= cos x e^bx/b - -sin x e^bx/b^2 - int - cos x e^bx/b^2

this is where i get stuck please help

Answer 1

Int Udv= uv -int vdu

let dv=cosx v=sinx
U=e^bx du=be^bx

int e^bx*cosx=e^bx(sinx)-int(bsinx*e^bx
now, to handel the last term, let
dv=sinx v=-cosx
u=e^bx du=be^bx
int e^bx coxx=e^bx sinx-b[-e^bx cosx -int(-cosx be^bx)
int e^bx cosx=e^bx sinx+be^bx cos x+b^2e^bx cosx
gathering terms,
(1-b^2)int e^bx cosx=e^bx(sinx+bcosx)

and you can solve for your original integral underlined. CHECK MY WORK