Problem: Evaluate the integral

Solution: This is a perfect example of an integral from the box "integrals with roots". It says there that we should get rid of the roots by substituting for 2x − 4 a suitable power. The lower the better, the least possible is given by the least common mutiple of all the roots; in our case this means 12:

Now we face the partial fractions decomposition, since we note that no smart substitution would help. First we do the long division, in the remainder we then factor the denominator. This is hard in general, but fortunately we can guess one root y = −1. Then we do partial fractions decomposition.

The quadratic term cannot be further factored. We found the constant at the linear term using the cover-up method, the remaining two for instance by the plug-in trick, for details see here. Now we can integrate. The first four terms are easy, only the last one requires more work. According to algorithm it is necessary to first complete the square in the denominator and use substitution to make the linear part disappear, then split the fraction in two and the rest is clear.

Now we put it together and then do the back substitution. One notable step is to remove the number 4/3 from logarithm and then merge it into the constant C.


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