Question

Integrate (e^cos5x) dx

Answer 1

Use integretation by parts 

=Scos(5x)e^x 
let u = cos(5x) du = -sin(5x)/5 
dv = e^x v = e^x 

=uv-Svdu 
=cos(5x)*e^x +(1/5) Se^x* sin(5x) 

Do integration by parts again 
let u=sin(5x) du= cos(5x)/5 and dv=e^x v=e^x 
Scos(5x)e^x =cos(5x)*e^x + (1/5)[sin(5x)e^x-(1/5)Se^xcos(5x)] 
Since the integral is similar to the starting equation we just move it to the other side so... 

(26/25)Scos(5x)e^x = cos(5x)e^x + (1/5)sin(5x)e^x 
and finally 
integral of cos(5x)e^x = (25/26)cos(5x)e^x + (5/26)sin(5x)e^x

Answer 2

integral [cos5x e^x] dx 
= uv - integral(v du) dx 
= cos(5x) e^x - integral [e^x * -5sin(5x)]dx 
= cos(5x) e^x + 5 integral [e^x * sin(5x)]dx 
= cos(5x) e^x + 5 {sin5x e^x - integral[e^x * 5cos5x]dx} 
= cos5x e^x + 5sin5x e^x - 5integral[5cos5x e^x]dx 
= cos5x e^x + 5sin5x e^x - 25integral[cos5x e^x]dx 
~~~ 
26 integral [cos5x e^x] dx = cos5x e^x + 5sin5x e^x 
integral [cos5x e^x] dx = [(e^x)/26]*(cos5x + 5sin5x)