integral from 1 to 3 of x/(sqrt(4x-3))

解答

\int_{1}^{3} \frac{x}{4 x - 3} d x

\frac{11}{6}

十进制

1.83333 \dots

求解步骤

\int_{1}^{3} \frac{x}{4 x - 3} d x

使用换元积分法

= \int_{1}^{3} \frac{u ^{2} + 3}{8} d u

提出常数:

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

= \frac{1}{8} \cdot \int_{1}^{3} u^{2} + 3 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}{8} (\int_{1}^{3} u^{2} d u + \int_{1}^{3} 3 d u)

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

\int_{1}^{3} 3 d u = 6

= \frac{1}{8} (\frac{26}{3} + 6)

化简

\frac{1}{8} (\frac{26}{3} + 6) : \frac{11}{6}

= \frac{11}{6}

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