integral from-1 to 1 of x/((sqrt(5-4x)))

解答

\int_{- 1}^{1} \frac{x}{( 5 - 4 x )} d x

\frac{1}{6}

十进制

0.16666 \dots

求解步骤

\int_{- 1}^{1} \frac{x}{5 - 4 x} d x

使用换元积分法

= \int_{3}^{1} \frac{u ^{2} - 5}{8} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{1}^{3} \frac{u ^{2} - 5}{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} - 5 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} 5 d u)

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

\int_{1}^{3} 5 d u = 10

= - \frac{1}{8} (\frac{26}{3} - 10)

化简

- \frac{1}{8} (\frac{26}{3} - 10) : \frac{1}{6}

= \frac{1}{6}

\int_{4}^{2} 2 x d x

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