Evaluate:`int(x^2+1)/(x^2-5x+6)dx`

1.	Evaluate:`int(x^2+1)/(x^2-5x+6)dx`
Answer» Here the integrand is not a proper rational function , on dividing `(x^(2)+1)"by" (x^(2)-5x+6),`we get `((x^(2)+1))/((x^(2)-5x+6))=1+((5x-5))/((x^(2)-5x+6))=1+((5x-5))/((x-2)(x-3)).` `Now ,Let ((5x-5))/((x-2)(x-3))=(A)/((x-2))+(B)/((x-3))` `implies ((5x-5))/((x-2)(x-3))=(A(x-3)+B(x-2))/((x-2)(x-3))` `implies (5x-5-=A(x-3)+B(x-2).` Putting `x=2` on both sides of (i) we get `A=-5.` Putting `x=3` on both sides of (i) , we get `B=10.` `therefore ((x^(2)+1))/((x^(2)-5x+6))=1-(5)/((x-2))+(10)/((x-3))` `implies int((x^(2)+1))/((x^(2)-5x+6))dx=intdx-5int(dx)/((x-2))+10int(dx)/((x-3))` `=x-5 log \|x-2\|+10 log \|x-3\|+c.`

Discussion

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