integral from 0 to 20 of x/(sqrt(9+2x))

解答

\int_{0}^{20} \frac{x}{9 + 2 x} d x

\frac{104}{3}

十进制

34.66666 \dots

求解步骤

\int_{0}^{20} \frac{x}{9 + 2 x} d x

使用换元积分法

= \int_{3}^{7} \frac{u ^{2} - 9}{2} d u

提出常数:

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

= \frac{1}{2} \cdot \int_{3}^{7} u^{2} - 9 d u

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int_{3}^{7} u^{2} d u - \int_{3}^{7} 9 d u)

\int_{3}^{7} u^{2} d u = \frac{316}{3}

\int_{3}^{7} 9 d u = 36

= \frac{1}{2} (\frac{316}{3} - 36)

化简

\frac{1}{2} (\frac{316}{3} - 36) : \frac{104}{3}

= \frac{104}{3}

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