Question

Evaluate the integral by using substitution.

`int_1^2 (1/x- 1/(2x^2))e^(2x) dx`

Answer 1

Let `I = int_1^2 e^(2x) (1/x - 1/(2x^2))  dx`

Put 2x = t

⇒ 2dx = dt

When x = 1, t = 2

And when x = 2, t = 4

∴ `I = 1/2 int_2^4 e^t (2/t - (1 xx4)/(2t^2))  dt`

`= 1/2 int_2^4 e^t (2/t - 2/t^2) dt`

`= int_2^4 e^t* (1/t - 1/t^2) dt`

`= int_2^4 e^t *[1/t + d/dt (1/t)] dt`

`= [e^t * 1/t]_2^4 = 1/4 e^4 - e^2/2`

`= e^2/2 (e^2/2 - 1)`

or `(e^2 (e^2 - 2))/4`