Partial Fraction Decomposition Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

medium
Integrate ∫1x2−4 dx\displaystyle\int \frac{1}{x^2-4}\,dx.

Solution

  1. 1
    (x−2)(x+2)(x-2)(x+2): A/(x−2)+B/(x+2)A/(x-2)+B/(x+2). A=1/4A=1/4, B=−1/4B=-1/4.
  2. 2
    14ln⁡∣x−2∣−14ln⁡∣x+2∣+C=14ln⁡∣x−2x+2∣+C\frac{1}{4}\ln|x-2|-\frac{1}{4}\ln|x+2|+C = \frac{1}{4}\ln\left|\frac{x-2}{x+2}\right|+C.

Answer

14ln⁡∣x−2x+2∣+C\frac{1}{4}\ln\left|\frac{x-2}{x+2}\right| + C
Factor the difference of squares, apply partial fractions, integrate logarithmically.

About Partial Fraction Decomposition

Breaking a rational expression into a sum of simpler fractions whose denominators are the factors of the original denominator.

Learn more about Partial Fraction Decomposition →

More Partial Fraction Decomposition Examples

Example 1 easy

Decompose [formula] and integrate.

Example 2 hard

Integrate [formula].

Example 3 easy

Decompose [formula].

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