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

1 Answer

May 19, 2015

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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