Example 7.2.12(c): Integrating Rational Functions
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Find the integral
dx
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This time it seems that we can not use partial fraction decomposition because
the degree of the numerator is higher than that of the denominator. To
remedy that problem, we use "long division" to divide the polynomials:
= x + 2 +
Integrating
x + 2 is easy, so the problem is reduced to finding
dx
and the partial fractions decomposition theorem applies just fine to this
integrand. We know that
Therefore we get three equations in three unknowns:
-3B + 2C = 12 | (for the constant coefficient) |
-3A + B = 2 | (for the x coefficient) |
A + C = 4 | (for the x2 coefficient) |
Solving this system of equations gives
A = -10/11, B = -8/11, C = 54/11
Therefore the one complicated integral above changes into three simpler ones:
The second integral is easy (involving the
ln). The first integral
seems to be somewhat difficult, because
arctan(x) = 1 / (1 + x2)
which does not quite work, and the numerator is
not the derivative of the
denominator, so the
ln is out, too. But a little algebra and some
substitution will do the trick:
Taking everything together gives: