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Evaluate the integral: integral (dx)/(1-x2)

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Question 9

Evaluate the integral:

∫dx1−x2\int \frac{dx}{1-x^2}

  1. Factor the denominator.

    The denominator is a difference of squares: 1−x2=(1−x)(1+x)1-x^2 = (1-x)(1+x).

  2. Perform partial fraction decomposition.

    Let 1(1−x)(1+x)=A1−x+B1+x\frac{1}{(1-x)(1+x)} = \frac{A}{1-x} + \frac{B}{1+x}.
    Multiply by (1−x)(1+x)(1-x)(1+x):
    1=A(1+x)+B(1−x)1 = A(1+x) + B(1-x)
    Set x=1x=1: $1 = A(1

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