Question

\(\displaystyle \int \frac{x+2}{x^{2}-4}\,dx=\) ?

• A. \(\log|x+2|+k\)
• B. \(\log|x^{2}-4|+k\)
• C. \(\log|x-2|+k\)
• D. \(\log\left|\frac{x+2}{x-2}\right|+k\)

Hint:

If the numerator equals a factor of the denominator, partial fractions may collapse to a single term.

Answer 1

The Correct Option is C

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 \[ \int \frac{x+2}{x^{2}-4}dx=\int \frac{1}{x-2}dx=\log|x-2|+k. \]