Question

Integrals using U sub. Please show steps so I can learn too :)

Answer 1

`(arctan^2(x))/(2)`

Explanation:

For this question, set u = arctan(x). This would be the easiest way because the derivative of arctan(x) is
`(1)/(x^2+1)` which is what we have. Before setting u, always look at the question and think about possible derivatives.

`\int(arctanx)/(x^2+1)dx \ Let \ u = arctan(x)\\\\\\u = arctan(x) \ \ \ du = (1)/(x^2+1)dx\\`

Next, plug in u and du.

`\int (arctan(x)*(1)/(x^2+1))dx` (rearranging)

`\int u\ du`

`= (u^2)/(2) = (1)/(2)u^2`

Substitute arctan(x) back in for u and add +C.

`(arctan^2(x))/(2)+C`