Indefinite Integration
Integration of Rational Functions
Grade 12
Question:
<p>\(\displaystyle\int\frac{dx}{(x-\alpha)(x-\beta)}\) equals \((\alpha\neq\beta)\)</p>
<li>\(\dfrac{1}{\alpha-\beta}\ln\!\left|\dfrac{x-\alpha}{x-\beta}\right|+C\)</li>
<li>\(\dfrac{1}{\beta-\alpha}\ln\!\left|\dfrac{x-\beta}{x-\alpha}\right|+C\)</li>
<li>\(\ln|(x-\alpha)(x-\beta)|+C\)</li>
<li>\(\dfrac{1}{\alpha-\beta}\ln\!\left|\dfrac{x-\alpha}{x-\beta}\right|+C\)</li>
Step-by-Step Solution
Key Concept: Partial fractions: 1/((x-\alpha)(x-\beta)) = (1/(\alpha-\beta)) \cdot [1/(x-\alpha) - 1/(x-\beta)]. Integrate each term.
<p><strong>Partial fractions:</strong></p>
<p>\[\frac{1}{(x-\alpha)(x-\beta)} = \frac{1}{\alpha-\beta}\left(\frac{1}{x-\alpha}-\frac{1}{x-\beta}\right)\]</p>
<p>\[\Rightarrow\int = \frac{1}{\alpha-\beta}\ln\!\left|\frac{x-\alpha}{x-\beta}\right|+C\]</p>
<p>Answer: <strong>(D)</strong></p>
Correct Answer: D