integral from 50 to 100 of x/(16+8x^2)

解答

\int_{50}^{100} \frac{x}{16 + 8 x ^{2}} d x

\frac{1}{16} ln (\frac{1667}{417})

十进制

0.08660 \dots

求解步骤

\int_{50}^{100} \frac{x}{16 + 8 x ^{2}} d x

使用换元积分法

= \int_{20016}^{80016} \frac{1}{16 u} d u

提出常数:

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

= \frac{1}{16} \cdot \int_{20016}^{80016} \frac{1}{u} d u

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{16} [ln ∣ u ∣]_{20016}^{80016}

带入上下限计算后得:

ln (80016) - ln (20016)

= \frac{1}{16} (ln (80016) - ln (20016))

化简

= \frac{1}{16} ln (\frac{1667}{417})

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