Question

If intgrate root 2x+3/x^2-5x+6dx=Aln|x-3|+Bln|x-2|+C then A+B=?

Answer 1

HELLO DEAR,




GIVEN:- $\bold{\int{\frac{2x + 3}{x^2 - 5x + 6}\,dx}}$

solving by partial fraction,


(2x + 3)/(x² - 5x + 6) = (2x + 3)/(x² - 2x - 3x + 6)

=> (2x + 3)/(x - 2)(x - 3) = A'/(x - 2) + B'/(x - 3)

=> 2x + 3 = A'(x - 3) + B'(x - 2)

=> 2x + 3 = x(A' + B') + (-3A' - 2B')

ok comparing both side like power of x,

A' + B' = 2 , 3A' + 2B' = -3

on solving we get,

A' = -7 , B' = 9


now, $\bold{\int{-7/(x - 2) + 9/(x - 3)}\,dx}$

$\bold{\implies 9log|x - 3| - 7log|x - 2| + c.}$



on comparing LHS , RHS we get,


A = 9 , B = -7

so, A + B = 9 - 7 = 2





I HOPE IT'S HELP YOU DEAR,
THANKS