integral from 4 to 6 of arctan(4/x)

解答

\int_{4}^{6} arctan (\frac{4}{x}) d x

6 arctan (\frac{2}{3}) + ln (\frac{169}{64}) - π

十进制

1.35743 \dots

求解步骤

\int_{4}^{6} arctan (\frac{4}{x}) d x

使用分布积分法

= [x arctan (\frac{4}{x}) - \int - \frac{4 x}{16 + x ^{2}} d x]_{4}^{6}

\int - \frac{4 x}{16 + x ^{2}} d x = - 2 ln 16 + x^{2}

= [x arctan (\frac{4}{x}) - (- 2 ln 16 + x^{2})]_{4}^{6}

化简

= [x arctan (\frac{4}{x}) + 2 ln 16 + x^{2}]_{4}^{6}

带入上下限计算后得:

6 arctan (\frac{2}{3}) + 2 ln (52) - π - 10 ln (2)

= 6 arctan (\frac{2}{3}) + 2 ln (52) - π - 10 ln (2)

化简

= 6 arctan (\frac{2}{3}) + ln (\frac{169}{64}) - π

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