Find : ∫ ((2x-5)e2x)/(2x-3)3 dx

∫ ((2x-5)e2x)/(2x-3)3 dx
Put 2x = t , dx = dt/2 , we get
∫ ((t-5)et)/(t-3)3 dt/2
Now observe that
(t-5)/(t-3)3 = 1/(t-3)2 - 2/(t-3)3
so we get
(1/(t-3)2 - 2/(t-3)3)et dt/2
if we consider f(t) = 1/(t-3)2 , then f'(t) = -2/(t-3)3
So the given problem is reduced to
∫(f(t)+f'(t))et dt/2 = f(t)et/2 + c= 1/(t-3)2 et/2 + c , now substitute t=2x , we get
1/(2x-3)2 e2x/2 + c