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受欢迎的微积分 >

integral of 1/(-196.2-4v^2)

解答

\int \frac{1}{- 196.2 - 4 v ^{2}} d v

解答

- 0.03569 \dots arctan (0.14278 \dots v) + C

求解步骤

\int \frac{1}{- 196.2 - 4 v ^{2}} d v

因式分解

- 196.2 - 4 v^{2} : - (196.2 + 4 v^{2})

= \int \frac{1}{- ( 196.2 + 4 v ^{2} )} d v

提出常数:

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

= - \int \frac{1}{196.2 + 4 v ^{2}} d v

使用换元积分法

= - \int \frac{7.00357 \dots}{196.2 + 196.2 u ^{2}} d u

提出常数:

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

= - 7.00357 \dots \cdot \int \frac{1}{196.2 + 196.2 u ^{2}} d u

196.2 + 196.2 u^{2} = 196.2 (1 + \frac{196.2 u ^{2}}{196.2})

= - 7.00357 \dots \cdot \int \frac{1}{196.2 ( 1 + \frac{196.2 u ^{2}}{196.2} )} d u

提出常数:

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

= - 7.00357 \dots \cdot \frac{1}{196.2} \cdot \int \frac{1}{1 + \frac{196.2 u ^{2}}{196.2}} d u

化简

= - 7.00357 \dots \cdot 0.00509 \dots \cdot \int \frac{1}{1 + \frac{196.2 u ^{2}}{196.2}} d u

约分:

196.2

= - 7.00357 \dots \cdot 0.00509 \dots \cdot \int \frac{1}{1 + u ^{2}} d u

使用积分常见定则:

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

= - 7.00357 \dots \cdot 0.00509 \dots arctan (u)

u = \frac{2}{14.00714 \dots} v

代回

= - 7.00357 \dots \cdot 0.00509 \dots arctan (\frac{2}{14.00714 \dots} v)

化简

= - 0.03569 \dots arctan (0.14278 \dots v)

解答补常数

= - 0.03569 \dots arctan (0.14278 \dots v) + C

作图

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