Question

How do you find the integral of `(arctan(2x)) / (1+4x^2)`?

Answer 1

Do a substitution: `u=arctan(2x), du=1/(1+(2x)^2)\cdot 2\ dx=2/(1+4x^2)\ dx`, giving

`\int arctan(2x)/(1+4x^2)\ dx=\frac{1}{2}\int u\ du=\frac{1}{4}u^{2}+C`

Hence,

`\int arctan(2x)/(1+4x^2)\ dx=\frac{1}{4}arctan^{2}(2x)+C`.