integral of 1/(4-v^2)

解答

\int \frac{1}{4 - v ^{2}} d v

\frac{1}{4} (ln \frac{v}{2} + 1 - ln \frac{v}{2} - 1) + C

求解步骤

\int \frac{1}{4 - v ^{2}} d v

使用换元积分法

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

提出常数:

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

= \frac{1}{2} \cdot \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}

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

u = \frac{v}{2}

代回

= \frac{1}{2} (\frac{ln \frac{v}{2} + 1}{2} - \frac{ln \frac{v}{2} - 1}{2})

化简

\frac{1}{2} (\frac{ln \frac{v}{2} + 1}{2} - \frac{ln \frac{v}{2} - 1}{2}) : \frac{1}{4} (ln \frac{v}{2} + 1 - ln \frac{v}{2} - 1)

= \frac{1}{4} (ln \frac{v}{2} + 1 - ln \frac{v}{2} - 1)

解答补常数

= \frac{1}{4} (ln \frac{v}{2} + 1 - ln \frac{v}{2} - 1) + C

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