What is the integral of 4arctan(2x) ?

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

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integral of 4arctan(2x)

\int 4 arctan (2 x) d x

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

Solution steps

\int 4 arctan (2 x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int arctan (2 x) d x

Apply Integration By Parts

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

= 4 (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

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

Simplify

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

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

Add a constant to the solution

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

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