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受欢迎的微积分 >

integral of (sqrt(x+16))/x

解答

\int \frac{x + 16}{x} d x

解答

- 8 (\frac{1}{2} ln \frac{x + 16}{4} + 1 - \frac{1}{2} ln \frac{x + 16}{4} - 1 - \frac{1}{4} x + 16) + C

求解步骤

\int \frac{x + 16}{x} d x

使用换元积分法

= \int \frac{2 u ^{2}}{u ^{2} - 16} d u

提出常数:

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

= 2 \cdot \int \frac{u ^{2}}{u ^{2} - 16} d u

因式分解

u^{2} - 16 : - (- u^{2} + 16)

= 2 \cdot \int \frac{u ^{2}}{- ( - u ^{2} + 16 )} d u

提出常数:

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

= 2 (- \int \frac{u ^{2}}{- u ^{2} + 16} d u)

使用换元积分法

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

提出常数:

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

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

\frac{v ^{2}}{- v ^{2} + 1} = \frac{1}{- v ^{2} + 1} - 1

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

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int 1 d v = v

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

代回

= 2 - 4 \frac{ln \frac{x + 16}{4} + 1}{2} - \frac{ln \frac{x + 16}{4} - 1}{2} - \frac{x + 16}{4}

化简

2 - 4 \frac{ln \frac{x + 16}{4} + 1}{2} - \frac{ln \frac{x + 16}{4} - 1}{2} - \frac{x + 16}{4} : - 8 (\frac{1}{2} ln \frac{x + 16}{4} + 1 - \frac{1}{2} ln \frac{x + 16}{4} - 1 - \frac{1}{4} x + 16)

= - 8 (\frac{1}{2} ln \frac{x + 16}{4} + 1 - \frac{1}{2} ln \frac{x + 16}{4} - 1 - \frac{1}{4} x + 16)

解答补常数

= - 8 (\frac{1}{2} ln \frac{x + 16}{4} + 1 - \frac{1}{2} ln \frac{x + 16}{4} - 1 - \frac{1}{4} x + 16) + C

作图

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