how to represent $\int \frac{\arctan \left(x\right)}{x}dx$ as a power series?

Question

I have no idea. I don t even no how to calculate the primitive can you help me?

Answer 1

1. Start with the Maclaurin series for arctangent, found here.

2. Divide term-by-term by $x$.

3. Integrate term-by-term.

Answer 2

Hint: You are trying to find an antiderivative for $\frac{1}{x}\arctan x$ and then find a power series for the antiderivative. What about if you try and do these steps in the other order?

Answer 3

I don't even no how to calculate the primitive

Nobody does. :-) It can be proven, using either Liouville's theorem or the Risch algorithm, that its antiderivative cannot be expressed in terms of elementary functions. In fact, one would need poly-logarithms to do that. But a closed-form expression for its primitive is not needed in order to find its power series: see Vadim's answer.