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受欢迎的微积分 >

integral of (3x)/(6x^2+9x)

解答

\int \frac{3 x}{6 x ^{2} + 9 x} d x

解答

\frac{1}{2} ln ∣ 2 x + 3 ∣ + C

求解步骤

\int \frac{3 x}{6 x ^{2} + 9 x} d x

提出常数:

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

= 3 \cdot \int \frac{x}{6 x ^{2} + 9 x} d x

化简

\frac{x}{6 x ^{2} + 9 x} : \frac{1}{3 ( 2 x + 3 )}

= 3 \cdot \int \frac{1}{3 ( 2 x + 3 )} d x

提出常数:

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

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

使用换元积分法

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

提出常数:

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

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

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 \cdot \frac{1}{3} \cdot \frac{1}{2} ln ∣ u ∣

u = 2 x + 3

代回

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

化简

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

= \frac{1}{2} ln ∣ 2 x + 3 ∣

解答补常数

= \frac{1}{2} ln ∣ 2 x + 3 ∣ + C

作图

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