Problem: Evaluate the integral

Solution: This integral fits the box "integration by parts", namely it is the type "logarithm multiplied by a power of x". We try the appropriate procedure:

We ended up with the same integral we started with, which is rather funny and does not happen too often. Fortunately the calculation was not futile, as the new integral has the opposite sign. This means that we can make an equation:

Is there any alternative? Yes it does and we should have thought of it right at the start. The derivative of logarithm is 1/x, which is something that we see next to dx. This suggests that the logarithmic substitution would succeed.

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