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Question
`int (2x + 1)/((x + 1) ( x - 2))` dx
Solution
Let I = `int (2x + 1)/((x + 1) ( x - 2))` dx
Using partial fraction
`(2x + 1)/((x + 1) (x -2)) = A/(x + 1) + B/(x - 2)`........(α)
2x + 1 = A (x - 2) + B( x + 1)
Put x = 2 in equation (i),
4 + 1 = B ( 2 + 1)
B = `5/3`
Put x = -1 in equation (i)
-2 + 1 = A[-1 -2]
A = `(-1)/(-3)`
A = `1/3`
Substituting the values of A and B in equation (α) we get
`(2x + 1)/((x + 1) (x -1)) = (1/3)/(x + 1) + (5/3)/(x -2)`
I = `int [(1/3)/(x +1) + (5/3)/( x -2)]`
I = `1/3 int 1/(x +1) dx + 5/3 int 1/(x -2) dx`
I = `1/3 log |x + 1| + 5/3 log |x -2| + c`
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