Question

Find : `int((2x-5)e^(2x))/(2x-3)^3dx`

Answer 1

Consider the given integral

`I=int((2x-5)e^(2x))/((2x-3)^2)dx`

Rewriting the above integral as

`I=inte^(2x-3) xxe^3(2x-3-2)/((2x-3)^3)dx`

`=e^3inte^(2x-3)[(2x-3)/(2x-3)^3-2/(2x-3)^3]dx`

`=e^3inte^(2x-3) [1/(2x-3)^2-2/(2x-3)^3]dx`

Let us consider, 2x -3 = t

⇒ 2dx = dt

`therefore I=e^3/2inte^t[(t-2)/t^3]dt`

Let `f(t)=1/t^2`

`f'(t)=(-2)/t^3`

if I = ∫e^t[f(t)+f'(t)]dt then, I = e^tf(t) + C

 `:.I=e^3/2xxe^txxf(t)+C`

`= e^3/2xxe^txx1/t^2+C`

`=e^3/2xxe^(2x-3)xx1/(2x-3)^2+C`

`=e^(2x)/(2(2x-3))+C`