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受欢迎的微积分 >

integral from 0 to 1 of ((3t^2)/(1+t^3))

解答

\int_{0}^{1} (\frac{3 t ^{2}}{1 + t ^{3}}) d t

解答

ln (2)

+1

十进制

0.69314 \dots

求解步骤

\int_{0}^{1} \frac{3 t ^{2}}{1 + t ^{3}} d t

提出常数:

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

= 3 \cdot \int_{0}^{1} \frac{t ^{2}}{1 + t ^{3}} d t

使用换元积分法

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

提出常数:

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

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

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 \cdot \frac{1}{3} [ln ∣ u ∣]_{1}^{2}

化简

3 \cdot \frac{1}{3} [ln ∣ u ∣]_{1}^{2} : [ln ∣ u ∣]_{1}^{2}

= [ln ∣ u ∣]_{1}^{2}

带入上下限计算后得:

ln (2)

= ln (2)

作图

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