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

解

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

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

解答ステップ

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

簡素化

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

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

部分積分を適用する

= - \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)

簡素化

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

定数を解答に追加する

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