Integration by Partial Fractions: Example 6

in mathematics •  2 months ago  (edited)

In this video I go over another example on integration of rational functions by partial fractions and this time go over an example that involves a non-linear function in the denominator that can't be factored any further. The example is on solving the integral of (4x2-3x+2)/(4x2-4x+3) and from this example we are able to obtain a general procedure for these types of integrals which end up being in the form of natural logs and inverse tangent functions.


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Integration of Rational Functions by Partial Fractions: Example 6

Integration by Partial Fractions Example 6.jpeg

Example:

Solution:

Thus we can't reduce further.

We also can't use partial fraction decomposition because we will get back to the same place we started, for example:

We can try completing the square instead:

Note: This example illustrates the general procedure for integrating a partial fraction of the form:

We complete the square in the denominator and then make a substitution that brings the integral into the form:

Then the first integral is a logarithm and the second is expressed in terms of tan-1.

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