Find \int \frac{x^2+1}{x^2-5x+6} dx.

(a) x – 5log|x-2| + 10log|x-3|+C

(b) x – 3log|x-2| + 5log|x-3|+C

(c) x – 10log|x-2| + 5log|x-3|+C

(d) x – 5log|x-5| + 10log|x-10|+C

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My enquiry is from Integration by Partial Fractions in portion Integrals of Mathematics – Class 12

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Correct choice is (a) x – 5log|x-2| + 10log|x-3|+C

Explanation: As it is not proper rational function, we divide numerator by denominator and get

\frac{x^2+1}{x^2-5x+6} = 1-\frac{5x-5}{x^2-5x+6} = 1+\frac{5x-5}{(x-2)(x-3)}

Let \frac{5x-5}{(x-2)(x-3)}=\frac{A}{(x-2)} + \frac{B}{(x-3)}

So that, 5x–5 = A(x-3) + B(x-2)

Now, equating coefficients of x and constant on both sides, we get A + B = 5 and 3A + 2B = 5. Solving these equations, we get A=-5 and B=10.

Therefore, \frac{x^2+1}{x^2-5x+6} = 1 – \frac{5}{(x-2)} + \frac{10}{(x-3)}.

\int \frac{x^2+1}{x^2-5x+6} dx = \int dx – 5\int \frac{dx}{(x-2)} + 10\int \frac{dx}{(x-3)}.

= x – 5log|x-2| + 10log|x-3|+C