Integrate.

Hint

You can find the derivative of arctan using the chain rule.

Answer

If you don't know the antiderivative of arctan *x* off the top of your head, that's to be expected. Thankfully there's a hidden factor of 1, so we can use integration by parts.Since we don't know the antiderivative of arctan *x* we can't use that for *v*', so we need to pick

*u* = arctan *x*

*v'* = 1

The derivative of arctan *x* is

which you could figure out using the chain rule, so

*u'* = (1 + *x*^{2})^{-1}

*v* = *x*.

Put everything in the formula:

Integrating the new integral by substitution (taking *u* = (1 + *x*^{2}) so *u*' = 2*x*) we get