integral from 1 to sqrt(3 of)(arctan(x))

解

\int_{1}^{3} (arctan (x)) d x

\frac{- 2 3 ln ( 2 ) + 4 π - 3 π}{4 3}

十進法表記

0.68182 \dots

解答ステップ

\int_{1}^{3} arctan (x) d x

部分積分を適用する

= [x arctan (x) - \int \frac{x}{x ^{2} + 1} d x]_{1}^{3}

\int \frac{x}{x ^{2} + 1} d x = \frac{1}{2} ln x^{2} + 1

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

簡素化

[x arctan (x) - \frac{1}{2} ln x^{2} + 1]_{1}^{3} : [\frac{1}{2} (2 x arctan (x) - ln x^{2} + 1)]_{1}^{3}

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

限度を計算する:

\frac{- 2 3 ln ( 2 ) + 4 π - 3 π}{4 3}

= \frac{- 2 3 ln ( 2 ) + 4 π - 3 π}{4 3}