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?

score 8 · Accepted Answer · answered Jun 11 '14 at 15:09

8

Start with the Maclaurin series for arctangent, found here.
Divide term-by-term by $x$.
Integrate term-by-term.

answered Jun 11 '14 at 15:09

vadim123

82,796

score 5 · Answer 2 · answered Jun 11 '14 at 15:10

5

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?

answered Jun 11 '14 at 15:10

Michael Albanese

99,526

score 2 · Answer 3 · answered Jun 11 '14 at 17:26

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.

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

3 Answers3