Integration of 1/(x² − 4) Using Partial Fractions

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Problem

Evaluate:

∫ dx / (x² − 4)

Solution

Given:

I = ∫ dx / (x² − 4)

First, factor the denominator:

x² − 4 = (x − 2)(x + 2)

Step 1: Partial Fraction Decomposition

1 / (x − 2)(x + 2) = A / (x − 2) + B / (x + 2)

1 = A(x + 2) + B(x − 2)

Step 2: Comparing Coefficients

1 = (A + B)x + (2A − 2B)

Comparing coefficients of x and constants:

A + B = 0 → A = −B

2A − 2B = 1

Substituting A = −B:

−2B − 2B = 1
−4B = 1
B = −1/4

Hence:

A = 1/4

Step 3: Substituting Values

1 / (x − 2)(x + 2) =

1/4 · 1/(x − 2) − 1/4 · 1/(x + 2)

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Step 4: Integrating

I = 1/4 ∫ dx / (x − 2) − 1/4 ∫ dx / (x + 2)

I = 1/4 · log|x − 2| − 1/4 · log|x + 2| + C

I = 1/4 · log | (x − 2)/(x + 2) | + C

This is the required result.