What is the integral of (x-arctan(x))/(x^2) ?

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

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integral of (x-arctan(x))/(x^2)

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

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

Solution steps

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

Expand

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

Simplify

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

Add a constant to the solution

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

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