Question

Find the value of \[ \int \frac{x}{(x-a)(x-b)(x-c)} \, dx. \]

Hint:

Partial fractions simplify rational functions into easily integrable terms.

Answer 1

Thegivenintegralcanbesolvedusingpartialfractiondecomposition: \[ \frac{x}{(x-a)(x-b)(x-c)}=\frac{A}{x-a}+\frac{B}{x-b}+\frac{C}{x-c}. \] Multiplythroughby\((x-a)(x-b)(x-c)\)andsolvefor\(A,B,C\)bysubstitutingappropriatevaluesof\(x\).Afterfindingthecoefficients,integrateeachtermindividually: \[ \int\frac{A}{x-a}\,dx+\int\frac{B}{x-b}\,dx+\int\frac{C}{x-c}\,dx. \] Theresultis: \[ \ln|x-a|,\ln|x-b|,\ln|x-c|, \] combinedwiththecoefficientstogivethefinalsolution.