The Closed Form Still Exists!

We know that the geometric progression sum of

$1 - x + x^2 - x^3 + x^4 - \cdots$

can be written as $\dfrac1{1+x}$ , where $-1<x<1$ .

If we integrate each of these terms with respect to $x$ , we get the series below. Which of the following is equivalent to the series below?

$x -\dfrac12 x^2 + \dfrac13x^3 - \dfrac14 x^4 + \dfrac15x^5 - \cdots$

Assume we ignore the arbitrary constant of integration .