Question

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

Answer 1

Answer:

$\frac{arctan^2(x)}{2}$

Step-by-step explanation:

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

$\int\frac{arctanx}{x^2+1}dx \ Let \ u = arctan(x)\\\\\\u = arctan(x) \ \ \ du = \frac{1}{x^2+1}dx$

Next, plug in u and du.

$\int (arctan(x)*\frac{1}{x^2+1})dx$ (rearranging)

$\int u\ du$

$= \frac{u^2}{2} = \frac{1}{2}u^2$

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

$\frac{arctan^2(x)}{2}+C$