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受欢迎的微积分 >

integral from 0 to 2 of (2x)/(2+x^2)

解答

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

解答

ln (3)

+1

十进制

1.09861 \dots

求解步骤

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

提出常数:

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

= 2 \cdot \int_{0}^{2} \frac{x}{2 + x ^{2}} d x

使用换元积分法

= 2 \cdot \int_{2}^{6} \frac{1}{2 u} d u

提出常数:

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

= 2 \cdot \frac{1}{2} \cdot \int_{2}^{6} \frac{1}{u} d u

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{2} [ln ∣ u ∣]_{2}^{6}

化简

2 \cdot \frac{1}{2} [ln ∣ u ∣]_{2}^{6} : [ln ∣ u ∣]_{2}^{6}

= [ln ∣ u ∣]_{2}^{6}

带入上下限计算后得:

ln (6) - ln (2)

= ln (6) - ln (2)

化简

= ln (3)

作图

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