Question:
Find the value of
\[
\int \frac{x}{(x-a)(x-b)(x-c)} \, dx.
\]
Find the value of
\[
\int \frac{x}{(x-a)(x-b)(x-c)} \, dx.
\]
Show Hint
Partial fractions simplify rational functions into easily integrable terms.
Updated On: Mar 1, 2025
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Solution and Explanation
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.
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