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受欢迎的微积分 >

integral of 5/(4-3x)

解答

\int \frac{5}{4 - 3 x} d x

解答

- \frac{5}{3} ln ∣ 4 - 3 x ∣ + C

求解步骤

\int \frac{5}{4 - 3 x} 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}{4 - 3 x} 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 (- \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 (- \frac{1}{3} ln ∣ u ∣)

u = 4 - 3 x

代回

= 5 (- \frac{1}{3} ln ∣ 4 - 3 x ∣)

化简

5 (- \frac{1}{3} ln ∣ 4 - 3 x ∣) : - \frac{5}{3} ln ∣ 4 - 3 x ∣

= - \frac{5}{3} ln ∣ 4 - 3 x ∣

解答补常数

= - \frac{5}{3} ln ∣ 4 - 3 x ∣ + C

作图

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(dx)/(dt)=0.6-(5x)/(1000+t)

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