integral from 0 to 1/5 of arctan(5x)

解答

\int_{0}^{\frac{1}{5}} arctan (5 x) d x

\frac{π - 2 ln ( 2 )}{20}

十进制

0.08776 \dots

求解步骤

\int_{0}^{\frac{1}{5}} arctan (5 x) d x

使用分布积分法

= [x arctan (5 x) - \int \frac{5 x}{25 x ^{2} + 1} d x]_{0}^{\frac{1}{5}}

\int \frac{5 x}{25 x ^{2} + 1} d x = \frac{1}{10} ln 25 x^{2} + 1

= [x arctan (5 x) - \frac{1}{10} ln 25 x^{2} + 1]_{0}^{\frac{1}{5}}

化简

[x arctan (5 x) - \frac{1}{10} ln 25 x^{2} + 1]_{0}^{\frac{1}{5}} : [\frac{1}{10} (10 x arctan (5 x) - ln 25 x^{2} + 1)]_{0}^{\frac{1}{5}}

= [\frac{1}{10} (10 x arctan (5 x) - ln 25 x^{2} + 1)]_{0}^{\frac{1}{5}}

带入上下限计算后得:

\frac{π - 2 ln ( 2 )}{20}

= \frac{π - 2 ln ( 2 )}{20}

\int_{\frac{1}{2}}^{1} \frac{3 y + 4}{y ^{2} + y} d y

\int_{6}^{10} (\frac{36}{x}) d x

\int_{- 1}^{1} x^{2} e^{- x} d x

\int_{- 3}^{3} (x^{2} + 6 x) d x

\int_{0}^{13} \frac{x}{( x + 2 ) ( x + 3 )} d x