Problem: Evaluate the integral

Solution: This is a typical integral from the box "integration by parts", namely the type "removing logarithm", so we apply the recommended method. We also notice that the integral is defined on the whole real line, so the given interval poses no problem. During the integration we then check that there will be no problem with points in this interval.

We obtained a rational function that we integrate using the usual procedure. First the long division, the remainder will lead to some arctangents or integrals solved by substitution (or both).

Thus we got the answer and it was not too difficult. However, a smart integrator could save some work. Indeed, the square in the logarithm can be eliminated by substitution at the very beginning. Does the substitution y = x2 + 1 stand a chance? For that we would need to find x next to dx as a derivative from this substitution. We do not have it there now, but we can easily create it there by borrowing one x from the cubic power. Then there will be only a square power left, which nicely ties in with our substitution. We are getting very optimistic. At the end we will integrate by parts anyway, but it will be much easier:

By the way, try to determine the indefinite integral, you should get

Remark: There is an interesting trick which can significantly shorten the first solution. Note that there is nothing forcing us to choose for g the simplest antiderivative, we can add a constant of our choice. In this example one can use it smartly.


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