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受欢迎的微积分 >

integral of (9x^2)/(x^3+8)

解答

\int \frac{9 x ^{2}}{x ^{3} + 8} d x

解答

3 ln x^{3} + 8 + C

求解步骤

\int \frac{9 x ^{2}}{x ^{3} + 8} d x

提出常数:

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

= 9 \cdot \int \frac{x ^{2}}{x ^{3} + 8} d x

使用换元积分法

= 9 \cdot \int \frac{1}{3 u} d u

提出常数:

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

= 9 \cdot \frac{1}{3} \cdot \int \frac{1}{u} d u

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 9 \cdot \frac{1}{3} ln ∣ u ∣

u = x^{3} + 8

代回

= 9 \cdot \frac{1}{3} ln x^{3} + 8

化简

9 \cdot \frac{1}{3} ln x^{3} + 8 : 3 ln x^{3} + 8

= 3 ln x^{3} + 8

解答补常数

= 3 ln x^{3} + 8 + C

作图

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