integral from 0 to 4 of (x/(sqrt(1+2x)))

解答

\int_{0}^{4} (\frac{x}{1 + 2 x}) d x

\frac{10}{3}

十进制

3.33333 \dots

求解步骤

\int_{0}^{4} \frac{x}{1 + 2 x} d x

使用换元积分法

= \int_{1}^{3} \frac{u ^{2} - 1}{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_{1}^{3} u^{2} - 1 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_{1}^{3} u^{2} d u - \int_{1}^{3} 1 d u)

\int_{1}^{3} u^{2} d u = \frac{26}{3}

\int_{1}^{3} 1 d u = 2

= \frac{1}{2} (\frac{26}{3} - 2)

化简

\frac{1}{2} (\frac{26}{3} - 2) : \frac{10}{3}

= \frac{10}{3}

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