Integrating x³/((x² + 1)(x - 1)²)

Calculus 2 · Axiom Academy

Partial fractions with a repeated linear factor and an irreducible quadratic, worked end to end. Evaluate the integral using partial fraction decomposition. Nicely done. You decomposed a rational function with two different factor types and integrated each piece on its own terms. Check the degree first: here , so the fraction is already proper — no long division needed. One term per factor type: the repeated linear factor (x-1)^2 contributes , and the irreducible quadratic x^2+1 contributes the linear numerator . Substitute then match: x = 1 isolates D instantly; comparing coefficients of like powers pins down the rest — here A = 0 , , C = 1 , . Each piece, its own rule: integrates to , to a logarithm, and by the power rule. The same recipe handles any rational function: factor the denominator completely, write the matching decomposition, solve for the coefficients, then integrate term by term.

This is the written version of the interactive lesson above. See the full Calculus 2 course.