Question

Find `int 1/((1+x^2)sqrt(1-arctanx)) dx`?

Answer 1

`int \ 1/((1+x^2)sqrt(1-arctanx)) \ dx = - 2sqrt(1-arctanx) + C`

Explanation:

We seek:

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

We can perform a substitution, Let:

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

Substituting into the integral we get:

`I = int \ (-1)/sqrt(u) \ du`
`\ \ = int \ (-1)/sqrt(u) \ du`
`\ \ = - \ int \ (1)/sqrt(u) \ du`

This is now a trivial integration, so sing the power rule:

`I = - 2sqrt(u) + C`

Finally, restoring the substitution:

`I = - 2sqrt(1-arctanx) + C`