What is the integral of arctan(x) 1/(x^2) ?

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

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integral of arctan(x) 1/(x^2)

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

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

Solution steps

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

Simplify

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

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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