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

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

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integral of 2x*arctan(x)

\int 2 x \cdot 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

= 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

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