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

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Answer 1 · 2015-05-19T20:50:31

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,

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

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