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受欢迎的微积分 >

integral of x/(x^4-4x^2+3)

解答

\int \frac{x}{x ^{4} - 4 x ^{2} + 3} d x

解答

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

求解步骤

\int \frac{x}{x ^{4} - 4 x ^{2} + 3} d x

使用换元积分法

= \int \frac{1}{4 u ( u + 3 - 4 u )} d u

提出常数:

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

= \frac{1}{4} \cdot \int \frac{1}{u ( u + 3 - 4 u )} d u

使用指数法则:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{u ( u + 3 - 4 u )} = u^{- \frac{1}{2}} \frac{1}{( u + 3 - 4 )}

= \frac{1}{4} \cdot \int u^{- \frac{1}{2}} \frac{1}{u + 3 - 4} d u

化简

u^{- \frac{1}{2}} \frac{1}{u + 3 - 4} : \frac{1}{u ( u - 1 )}

= \frac{1}{4} \cdot \int \frac{1}{u ( u - 1 )} d u

使用换元积分法

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

提出常数:

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

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

因式分解

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

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

提出常数:

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

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

使用积分常见定则:

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

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

代回

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

化简

\frac{1}{4} \cdot 2 - \frac{ln x ^{4} + 1}{2} - \frac{ln x ^{4} - 1}{2} : - \frac{1}{4} (ln x^{2} + 1 - ln x^{2} - 1)

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

解答补常数

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

作图

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