What is the integral of 2x^{-3}arctan(x) ?

The integral of 2x^{-3}arctan(x) is -(arctan(x))/(x^2)-1/x-arctan(x)+C

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integral of 2x^{-3}arctan(x)

\int 2 x^{- 3} arctan (x) d x

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

Solution steps

\int 2 x^{- 3} 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^{- 3} arctan (x) d x

Simplify

x^{- 3} arctan (x) : \frac{arctan ( x )}{x ^{3}}

= 2 \cdot \int \frac{arctan ( x )}{x ^{3}} d x

Apply Integration By Parts

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

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

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

Simplify

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

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

Add a constant to the solution

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

What is the integral of 2x^{-3}arctan(x) ?
The integral of 2x^{-3}arctan(x) is -(arctan(x))/(x^2)-1/x-arctan(x)+C