Integration by Partial Fractions
Problem
Evaluate the integral:
∫ dx / (x² − 4)
Solution
Let,
1 / (x² − 4) = 1 / [(x − 2)(x + 2)]
Now decompose the expression into partial fractions:
1 / [(x − 2)(x + 2)] = A / (x − 2) + B / (x + 2) ...(1)
Combining the fractions:
= [A(x + 2) + B(x − 2)] / (x² − 4)
= [(A + B)x + (2A − 2B)] / (x² − 4)
Comparing Coefficients
From both sides:
A + B = 0 ⇒ A = −B ...(2)
2A − 2B = 1 ...(3)
Substituting A = −B from equation (2) into equation (3):
−2B − 2B = 1
−4B = 1 ⇒ B = −1/4
Therefore:
A = 1/4
Substitute Values
1 / [(x − 2)(x + 2)] = (1/4)/(x − 2) − (1/4)/(x + 2)
Integrate Both Sides
∫ dx / (x² − 4) = (1/4) ∫ dx / (x − 2) − (1/4) ∫ dx / (x + 2)
= (1/4) ln|x − 2| − (1/4) ln|x + 2| + C
∴ I = (1/4) ln | (x − 2) / (x + 2) | + C