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 1/x is (up to the sign) equal to 1/x2, which is exactly what is available in the integral. Since the calculations seem to be quite complicated, we first find an antiderivative and then substitute the limits.

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 z = 2y − 1.

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.


Next problem
Back to Solved Problems - Integrals