integral from 2 to 3 of x^3arctan(x-1)

解答

\int_{2}^{3} x^{3} arctan (x - 1) d x

- π + \frac{8}{3} + \frac{- π + 85 arctan ( 2 ) - 24}{4}

十进制

16.26658 \dots

求解步骤

\int_{2}^{3} x^{3} arctan (x - 1) d x

使用分布积分法

= [\frac{1}{4} x^{4} arctan (x - 1) - \int \frac{x ^{4}}{4 ( x ^{2} - 2 x + 2 )} d x]_{2}^{3}

\int \frac{x ^{4}}{4 ( x ^{2} - 2 x + 2 )} d x = \frac{x ^{3}}{12} + \frac{1}{4} x^{2} + \frac{1}{2} x - arctan (x - 1)

= [\frac{1}{4} x^{4} arctan (x - 1) - (\frac{x ^{3}}{12} + \frac{1}{4} x^{2} + \frac{1}{2} x - arctan (x - 1))]_{2}^{3}

化简

= [\frac{1}{4} x^{4} arctan (x - 1) - \frac{x ^{3}}{12} - \frac{1}{4} x^{2} - \frac{1}{2} x + arctan (x - 1)]_{2}^{3}

带入上下限计算后得:

- π + \frac{8}{3} + \frac{- π + 85 arctan ( 2 ) - 24}{4}

= - π + \frac{8}{3} + \frac{- π + 85 arctan ( 2 ) - 24}{4}

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