Question

Evaluate \(\displaystyle\int\dfrac{x^2+1}{x^2-5x+6}\,dx\).

Hint:

Do polynomial long division first (equal degree top and bottom), then partial fractions on the remainder.

Answer 1

Step 1: Rewrite numerator relative to the denominator directly:
$x^2+1=(x^2-5x+6)+(5x-5)$ — added and subtracted to match the denominator exactly, confirming the quotient is 1 with remainder $5x-5$.

Step 2: Factor and split into partial fractions:
$(x-2)(x-3)$ in the denominator; solving $5x-5=A(x-3)+B(x-2)$ at the roots gives $A=-5$ (at $x=2$) and $B=10$ (at $x=3$).

Step 3: Integrate the three simple pieces:
$\int1\,dx=x$; $\int\frac{-5}{x-2}dx=-5\ln|x-2|$; $\int\frac{10}{x-3}dx=10\ln|x-3|$.

Final Answer:
\[ \boxed{x-5\ln|x-2|+10\ln|x-3|+C} \]