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

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

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

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

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

Solution steps

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

Simplify

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

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

Apply Integration By Parts

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

= - \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))

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

Simplify

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

Add a constant to the solution

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

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