What is the integral of-2xarctan(x) ?

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

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integral of-2xarctan(x)

\int - 2 x arctan (x) d x

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

Solution steps

\int - 2 x 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 x arctan (x) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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