Question

What is the antiderivative of `(1)/(1+x^2)`?

Answer 1

`arctan(x)+C`

Explanation:

This is one to memorize:

`\int 1/(1+x^2)\ dx=arctan(x)+C`

It can be derived by differentiation of both sides of the equation `tan(arctan(x))=x` (assuming you know that `d/dx(tan(x))=sec^{2}(x)`) and using the Chain Rule:

`sec^{2}(arctan(x))*d/dx(arctan(x))=1`

`\Rightarrow d/dx(arctan(x))=1/sec^[2}(arctan(x))`

`=1/(1+tan^{2}(arctan(x)))=1/(1+x^2)` for all `x\in RR`.