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

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

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integral of (x-arctan(x))/(x^3)

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

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

Solution steps

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

Expand

\frac{x - arctan ( x )}{x ^{3}} : \frac{1}{x ^{2}} - \frac{arctan ( x )}{x ^{3}}

= \int \frac{1}{x ^{2}} - \frac{arctan ( x )}{x ^{3}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{x ^{2}} d x - \int \frac{arctan ( x )}{x ^{3}} d x

\int \frac{1}{x ^{2}} d x = - \frac{1}{x}

\int \frac{arctan ( x )}{x ^{3}} d x = - \frac{arctan ( x )}{2 x ^{2}} + \frac{1}{2} (- \frac{1}{x} - arctan (x))

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

Simplify

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

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

Add a constant to the solution

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

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