integral of x*arctan(sqrt(x^2-1))

解

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

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

解答ステップ

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

部分積分を適用する

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

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

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

定数を解答に追加する

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