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受欢迎的微积分 >

integral of ((5x^{(2)}))/((2x^{(3))+1)}

解答

\int \frac{( 5 x ^{(2)} )}{( 2 x ^{(3)} + 1 )} d x

解答

\frac{5}{6} ln 2 x^{3} + 1 + C

求解步骤

\int \frac{5 x ^{2}}{2 x ^{3} + 1} d x

提出常数:

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

= 5 \cdot \int \frac{x ^{2}}{2 x ^{3} + 1} d x

使用换元积分法

= 5 \cdot \int \frac{1}{6 u} d u

提出常数:

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

= 5 \cdot \frac{1}{6} \cdot \int \frac{1}{u} d u

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 \cdot \frac{1}{6} ln ∣ u ∣

u = 2 x^{3} + 1

代回

= 5 \cdot \frac{1}{6} ln 2 x^{3} + 1

化简

5 \cdot \frac{1}{6} ln 2 x^{3} + 1 : \frac{5}{6} ln 2 x^{3} + 1

= \frac{5}{6} ln 2 x^{3} + 1

解答补常数

= \frac{5}{6} ln 2 x^{3} + 1 + C

作图

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