Q.Evaluate $\displaystyle\int \frac{x^2+1}{x(x-1)}\,dx$.

A proper rational function $\frac{f(x)}{g(x)}$ (degree of $f<$ degree of $g$) is decomposed into a sum of simpler fractions determined by how $g(x)$ factors: non-repeated linear factors give one constant-over-linear term per factor, $\frac{A}{x-a}+\frac{B}{x-b}+\cdots$; a repeated linear factor $(x-a)^2$ contributes two terms, $\frac{A}{x-a}+\frac{B}{(x-a)^2}$; and a non-repeated irreducible quadratic factor contributes a linear-over-quadratic term, $\frac{Bx+C}{x^2+bx+c}$. The unknown constants are found either by comparing coefficients of like powers of $x$ after clearing denominators, or (faster, for linear factors) by substituting the root of each factor in turn. An improper fraction (numerator degree $\geq$ denominator degree) must be reduced by polynomial long division first. Many integrals that do not look like ration …