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受欢迎的微积分 >

integral of 1/(x^{4/5)(1+x^{1/5})}

解答

\int \frac{1}{x ^{\frac{4}{5}} ( 1 + x ^{\frac{1}{5}} )} d x

解答

5 ln x^{\frac{1}{5}} + 1 + C

求解步骤

\int \frac{1}{x ^{\frac{4}{5}} ( 1 + x ^{\frac{1}{5}} )} d x

使用换元积分法

= \int \frac{5}{4 u ^{\frac{3}{4}} ( u ^{\frac{1}{4}} + 1 )} d u

提出常数:

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

= \frac{5}{4} \cdot \int \frac{1}{u ^{\frac{3}{4}} ( u ^{\frac{1}{4}} + 1 )} d u

使用换元积分法

= \frac{5}{4} \cdot \int \frac{4}{v} d v

提出常数:

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

= \frac{5}{4} \cdot 4 \cdot \int \frac{1}{v} d v

使用积分常见定则:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{5}{4} \cdot 4 ln ∣ v ∣

代回

= \frac{5}{4} \cdot 4 ln (x^{\frac{4}{5}})^{\frac{1}{4}} + 1

化简

\frac{5}{4} \cdot 4 ln (x^{\frac{4}{5}})^{\frac{1}{4}} + 1 : 5 ln x^{\frac{1}{5}} + 1

= 5 ln x^{\frac{1}{5}} + 1

解答补常数

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

作图

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