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受欢迎的微积分 >

integral of 1/(3sqrt(x)(4+\sqrt{x))}

解答

\int \frac{1}{3 x ( 4 + x )} d x

解答

\frac{2}{3} ln 4 + x + C

求解步骤

\int \frac{1}{3 x ( 4 + x )} d x

提出常数:

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

= \frac{1}{3} \cdot \int \frac{1}{x ( 4 + x )} d x

使用换元积分法

= \frac{1}{3} \cdot \int \frac{2}{u} d u

提出常数:

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

= \frac{1}{3} \cdot 2 \cdot \int \frac{1}{u} d u

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{3} \cdot 2 ln ∣ u ∣

u = 4 + x

代回

= \frac{1}{3} \cdot 2 ln 4 + x

化简

\frac{1}{3} \cdot 2 ln 4 + x : \frac{2}{3} ln 4 + x

= \frac{2}{3} ln 4 + x

解答补常数

= \frac{2}{3} ln 4 + x + C

作图

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