Question

How do you evaluate `\int \frac { 1} { ( 1+ x ^ { 2} ) \arctan ( x ) } d x`?

Answer 1

`int \ 1/((1+x^2)arctanx) \ dx = ln|arctan x| + c`

Explanation:

We want to find:

`I = int \ 1/((1+x^2)arctanx) \ dx`

We can perform a substitution: Let

`u = arctan x => (du)/dx = 1/(1+x^2)`

Applying the substitution to our integral gives:

`I = int \ 1/arctanx * 1/(1+x^2) \ dx`
`\ \ = int \ 1/u \ du`
`\ \ = ln|u| + c`

And restoring the substitution we get:

`I = ln|arctan x| + c`