integral from 0 to 9 of x/(sqrt(4+5x))

解答

\int_{0}^{9} \frac{x}{4 + 5 x} d x

\frac{22}{3}

十进制

7.33333 \dots

求解步骤

\int_{0}^{9} \frac{x}{4 + 5 x} d x

使用换元积分法

= \int_{2}^{7} \frac{2 ( u ^{2} - 4 )}{25} d u

提出常数:

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

= \frac{2}{25} \cdot \int_{2}^{7} u^{2} - 4 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{2}{25} (\int_{2}^{7} u^{2} d u - \int_{2}^{7} 4 d u)

\int_{2}^{7} u^{2} d u = \frac{335}{3}

\int_{2}^{7} 4 d u = 20

= \frac{2}{25} (\frac{335}{3} - 20)

化简

\frac{2}{25} (\frac{335}{3} - 20) : \frac{22}{3}

= \frac{22}{3}

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