Question:
\(\displaystyle \int \frac{x+2}{x^{2}-4}\,dx=\) ?
\(\displaystyle \int \frac{x+2}{x^{2}-4}\,dx=\) ?
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If the numerator equals a factor of the denominator, partial fractions may collapse to a single term.
Updated On: Oct 17, 2025
- \(\log|x+2|+k\)
- \(\log|x^{2}-4|+k\)
- \(\log|x-2|+k\)
- \(\log\left|\frac{x+2}{x-2}\right|+k\)
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The Correct Option is C
Solution and Explanation
Factor the denominator: \(x^{2}-4=(x-2)(x+2)\).
Partial fractions:
\[
\frac{x+2}{x^{2}-4}=\frac{A}{x-2}+\frac{B}{x+2} $\Rightarrow$ A(x+2)+B(x-2)=x+2.
\]
Compare coefficients: \(A+B=1\) and \(2A-2B=2\Rightarrow A-B=1\).
Solve: \(A=1,\ B=0\). Hence
\[
\int \frac{x+2}{x^{2}-4}dx=\int \frac{1}{x-2}dx=\log|x-2|+k.
\]
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