Thread: Math tricks for everyone View Single Post
P: 621
Integrand=$${[}{1}{-}{x}^{2}{+}{x}^{4}{-}{x}^{6}............{]}$$

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

$$\int\frac{1}{{1}{+}{x}^{2}}{dx}{=}{-}\int\frac{1}{{1}{+}{y}^{2}}{dy}$$

Since y<1 , we may proceed exactly in the same manner and get the same
result preceded by a negative sign as expected.

$${tan}^{-1}{(}{1}{/}{x}{)}{=}{\pi}{/}{2}{-}{tan}^{-1}{(}{x}{)}$$