Integrand=[tex]{[}{1}{-}{x}^{2}{+}{x}^{4}{-}{x}^{6}............{]}[/tex]

The above series is convergent for abs[x]<1. Integration is allowed for such cases.

When abs[x]>1 we may proceed as follows:

Let y=1/x

Now, abs value of y is less than 1

[tex]\int\frac{1}{{1}{+}{x}^{2}}{dx}{=}{-}\int\frac{1}{{1}{+}{y}^{2}}{dy}[/tex]

Since y<1 , we may proceed exactly in the same manner and get the same

result preceded by a negative sign as expected.

[tex]{tan}^{-1}{(}{1}{/}{x}{)}{=}{\pi}{/}{2}{-}{tan}^{-1}{(}{x}{)}[/tex]

Link:

http://en.wikipedia.org/wiki/Inverse...tric_functions