積分記号 arctan(x^2)

解

積分記号

arctan (x^{2})

x arctan (x^{2}) - 2 (- \frac{1}{4 2} (ln 2 x^{2} + 22 x + 2 - 2 arctan (2 x + 1)) + \frac{1}{4 2} (ln 2 x^{2} - 22 x + 2 + 2 arctan (2 x - 1))) + C

解答ステップ

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

部分積分を適用する

= x arctan (x^{2}) - \int \frac{2 x ^{2}}{x ^{4} + 1} d x

\int \frac{2 x ^{2}}{x ^{4} + 1} d x = 2 (- \frac{1}{4 2} (ln 2 x^{2} + 22 x + 2 - 2 arctan (2 x + 1)) + \frac{1}{4 2} (ln 2 x^{2} - 22 x + 2 + 2 arctan (2 x - 1)))

= x arctan (x^{2}) - 2 (- \frac{1}{4 2} (ln 2 x^{2} + 22 x + 2 - 2 arctan (2 x + 1)) + \frac{1}{4 2} (ln 2 x^{2} - 22 x + 2 + 2 arctan (2 x - 1)))

定数を解答に追加する

= x arctan (x^{2}) - 2 (- \frac{1}{4 2} (ln 2 x^{2} + 22 x + 2 - 2 arctan (2 x + 1)) + \frac{1}{4 2} (ln 2 x^{2} - 22 x + 2 + 2 arctan (2 x - 1))) + C