Question

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

The question was posed to me in an interview.

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

Answer 1

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