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

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##.

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