integral of (3arctan(x))/(x^2)

解

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

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

解答ステップ

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

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

部分積分を適用する

= 3 (- \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

= 3 (- \frac{arctan ( x )}{x} - (- ln ∣ x ∣ + \frac{1}{2} ln x^{2} + 1))

簡素化

= 3 (- \frac{arctan ( x )}{x} + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1)

定数を解答に追加する

= 3 (- \frac{arctan ( x )}{x} + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1) + C