13: If you decided on differentiation, you should get

This series has k + 1 in the denominator, by shifting index this can be changed into k, thus leading to one of the logarithmic expansion. If you do not remember them, you can also adjust the power and then use another differentiation to remove the k + 1.

Partial fractions go

These are again series leading to a logarithmic expansion, but the second series has to be adjusted first similarly to the first approach.

If you tried the third approach, you should get

Now it is time to integrate, it is necessary to use integration by parts. After integration there will be constant C, typically we would determine its value by substituting 0 into the series and the resulting expression on the right, but that is not possible here, since the calculation above only works for non-zero x. To circumvent it, apply limit for x going to 0 in that expression on the right to confirm that C = 0.

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