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受欢迎的微积分 >

integral of 5/(3x+1)

解答

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

解答

\frac{5}{3} ln ∣ 3 x + 1 ∣ + C

求解步骤

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

使用换元积分法

= 5 \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

= 5 \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 ∣)

= 5 \cdot \frac{1}{3} ln ∣ u ∣

u = 3 x + 1

代回

= 5 \cdot \frac{1}{3} ln ∣ 3 x + 1 ∣

化简

5 \cdot \frac{1}{3} ln ∣ 3 x + 1 ∣ : \frac{5}{3} ln ∣ 3 x + 1 ∣

= \frac{5}{3} ln ∣ 3 x + 1 ∣

解答补常数

= \frac{5}{3} ln ∣ 3 x + 1 ∣ + C

作图

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