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Example 7.2.12(b): Integrating Rational Functions

Find the integral (1 - x4)-1 dx
We want to again use partial fraction decomposition, so we need to factor the polynomial p(x) = 1 - x4. Clearly:
1 - x4 = (1 - x2) (1 + x2) = (1 + x) (1 - x) (1 + x2)
By the partial fraction decomposition theorem we therefore know that:
We can again combine the right side as follows:
so that
1 = A (1-x)(1+x2) + B(1+x)(1+x2) + (Cx+D)(1-x2) =
      = A + Ax2 - Ax - Ax3 + B + Bx2 + Bx + Bx3) + Cx - Cx3 + D - Dx2 =
      = (A + B + D) + x(-A + B + C) + x2(A + B - D) + x3(-A + B - C)
for all x. This gives four equations in four unknowns:
A + B + D= 1 (for the constant coefficient)
-A + B + C = 0 (for the x coefficient)
A + B - D = 0 (for the x2 coefficient)
-A + B - C = 0 (for the x3 coefficient)
which - perhaps after reviewing how to efficiently solve systems of linear equations - gives the answer:
A = 1/4, B = 1/4, D = 1/2, C = 0
Therefore we can solve our original integral
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