Question

Evaluate the integral: (5x+1)/((2x+1)(x-1))

Answer 1

( 5 x + 1 ) / ((2 x + 1 ) ( x - 1)) = A /(2 x + 1 ) + B /( x - 1 )
A ( x - 1 ) + B ( 2 x + 1 ) = A x - A + 2 B x + B
A x + 2 B x = 5 x
- A + B = 1
----------------------
A + 2 B = 5
- A + B = 1
------------------
3 B = 6
B = 6 : 3 = 2
A = 2 - 1 = 1
The integral becomes:

`\int { (dx)/(2x+1)} + \int { (2dx)/(x-1) }=`
= 1/2 ln ( 2 x + 1 ) + 2 ln ( x - 1 ) =
= ln √( 2 x + 1 ) + ln ( x - 1 )² =
= ln ((√2 x + 1) ( x - 1 )²) + C