Problem: Evaluate the integral
Solution:
This problem does not look like some standard type belonging to a box, so we
try to think of some substitution. We do not like those reciprocal x,
so we try to get rid of them. An experienced integrator would already see
that this substitution will succeed, because the derivative of
The integral now really looks better, it is time for partial fractions.
Because we have distinct linear factors, the unknown constants can be easily
obtained using the cover-up trick trick. If you want to see it, look
here.
Again, we simplify our job by not pulling out the two from the last factor.
The linear factor are then easily integrated using substitution, for
instance
Now we substitute the limits, note that we integrate over an interval that satisfies conditions of validity of this integral. The answer will not be very nice, but that's life.
We can also solve this integral in another way. We can get rid of the fractions in the denominator by multiplying each factor by x. We already have two of these available, the third one we easily create using the multiply-divide trick. This procedure leads to partial fractions, we will briefly outline the calculations:
Here you can find the details.
The answers are equal.