Integrate - x^2+1/x^2-5x+6 by using partial fraction method of integration.

Here the integrand is not a proper rational function ,so we divide x2+1 by x2-5x+6

x2+1/x2-5x+6 =1+(5x-5)/x2-5x+6 = 1+(5x-5)/(x-2)(x-3)

(5x-5)/(x-2)(x-3) = A/x-2+B/X-3

5x-5=A(x-3)+B(x-2)

equate the coefficients of constant terms and x on both sides ,you will get A+B=5 and 3A+2B=5 .Solve these equations then A=-5 AND B=10

(X2+1)/X2-5X+6 =1-5/X-2+10/X-3

∫(x2+1)/x2-5x+6 =∫dx-5∫1/x-2 dx+10∫dx/x-3

=x-5 log modulus(x-2) +10log(x-3)+c