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受欢迎的微积分 >

integral of 2/(3x^2-8)

解答

\int \frac{2}{3 x ^{2} - 8} d x

解答

- \frac{1}{2 6} (ln \frac{3}{2 2} x + 1 - ln \frac{3}{2 2} x - 1) + C

求解步骤

\int \frac{2}{3 x ^{2} - 8} d x

提出常数:

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

= 2 \cdot \int \frac{1}{3 x ^{2} - 8} d x

使用换元积分法

= 2 \cdot \int \frac{1}{2 6 ( u ^{2} - 1 )} d u

提出常数:

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

= 2 \cdot \frac{1}{2 6} \cdot \int \frac{1}{u ^{2} - 1} d u

因式分解

u^{2} - 1 : - (- u^{2} + 1)

= 2 \cdot \frac{1}{2 6} \cdot \int \frac{1}{- ( - u ^{2} + 1 )} d u

提出常数:

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

= 2 \cdot \frac{1}{2 6} (- \int \frac{1}{- u ^{2} + 1} d u)

使用积分常见定则:

\int \frac{1}{- u ^{2} + 1} d u = \frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}

= 2 \cdot \frac{1}{2 6} (- (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}))

u = \frac{3}{2 2} x

代回

= 2 \cdot \frac{1}{2 6} - \frac{ln \frac{3}{2 2} x + 1}{2} - \frac{ln \frac{3}{2 2} x - 1}{2}

化简

2 \cdot \frac{1}{2 6} - \frac{ln \frac{3}{2 2} x + 1}{2} - \frac{ln \frac{3}{2 2} x - 1}{2} : - \frac{1}{2 6} (ln \frac{3}{2 2} x + 1 - ln \frac{3}{2 2} x - 1)

= - \frac{1}{2 6} (ln \frac{3}{2 2} x + 1 - ln \frac{3}{2 2} x - 1)

解答补常数

= - \frac{1}{2 6} (ln \frac{3}{2 2} x + 1 - ln \frac{3}{2 2} x - 1) + C

作图

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