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QUESTION
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##.
