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

解

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

\frac{2 3 ln ( 2 ) + 2 π - 3 π}{4 3}

十進法表記

0.46807 \dots

解答ステップ

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

部分積分を適用する

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

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

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

簡素化

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

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

限度を計算する:

\frac{2 3 ln ( 2 ) + 2 π - 3 π}{4 3}

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