What is the integral of (arctan(2x)) ?

The integral of (arctan(2x)) is xarctan(2x)-1/4 ln|4x^2+1|+C

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integral of (arctan(2x))

\int (arctan (2 x)) d x

x arctan (2 x) - \frac{1}{4} ln 4 x^{2} + 1 + C

Solution steps

\int arctan (2 x) d x

Apply Integration By Parts

: x arctan (2 x) - \int \frac{2 x}{4 x ^{2} + 1} d x

= x arctan (2 x) - \int \frac{2 x}{4 x ^{2} + 1} d x

\int \frac{2 x}{4 x ^{2} + 1} d x = \frac{1}{4} ln 4 x^{2} + 1

= x arctan (2 x) - \frac{1}{4} ln 4 x^{2} + 1

Add a constant to the solution

= x arctan (2 x) - \frac{1}{4} ln 4 x^{2} + 1 + C

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What is the integral of (arctan(2x)) ?
The integral of (arctan(2x)) is xarctan(2x)-1/4 ln|4x^2+1|+C