What is the integral of 2arctan(x) ?

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

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integral of 2arctan(x)

\int 2 arctan (x) d x

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

Solution steps

\int 2 arctan (x) d x

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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