integral from 0 to sqrt(3 of)2xarctan(x)

解答

\int_{0}^{3} 2 x arctan (x) d x

\frac{4 π - 3 3}{3}

十进制

2.45673 \dots

求解步骤

\int_{0}^{3} 2 x arctan (x) d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int_{0}^{3} x arctan (x) d x

使用分布积分法

= 2 [\frac{1}{2} (x^{2} arctan (x) - 2 \cdot \int \frac{x ^{2}}{2 ( x ^{2} + 1 )} d x)]_{0}^{3}

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

= 2 [\frac{1}{2} (x^{2} arctan (x) - 2 \cdot \frac{1}{2} (- arctan (x) + x))]_{0}^{3}

化简

2 [\frac{1}{2} (x^{2} arctan (x) - 2 \cdot \frac{1}{2} (- arctan (x) + x))]_{0}^{3} : 2 [\frac{1}{2} (x^{2} arctan (x) + arctan (x) - x)]_{0}^{3}

= 2 [\frac{1}{2} (x^{2} arctan (x) + arctan (x) - x)]_{0}^{3}

带入上下限计算后得:

\frac{4 π - 3 3}{6}

= 2 \cdot \frac{4 π - 3 3}{6}

化简

= \frac{4 π - 3 3}{3}

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