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受欢迎的微积分 >

integral of 9/(x(x^4+8))

解答

\int \frac{9}{x ( x ^{4} + 8 )} d x

解答

\frac{9}{4} (\frac{1}{2} ln (x) - \frac{1}{8} ln x^{4} + 8) + C

求解步骤

\int \frac{9}{x ( x ^{4} + 8 )} d x

提出常数:

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

= 9 \cdot \int \frac{1}{x ( x ^{4} + 8 )} d x

使用换元积分法

= 9 \cdot \int \frac{1}{4 u ( u + 8 )} d u

提出常数:

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

= 9 \cdot \frac{1}{4} \cdot \int \frac{1}{u ( u + 8 )} d u

将

\frac{1}{u ( u + 8 )}

用部份分式展开:

\frac{1}{8 u} - \frac{1}{8 ( u + 8 )}

= 9 \cdot \frac{1}{4} \cdot \int \frac{1}{8 u} - \frac{1}{8 ( u + 8 )} d u

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

= 9 \cdot \frac{1}{4} (\int \frac{1}{8 u} d u - \int \frac{1}{8 ( u + 8 )} d u)

\int \frac{1}{8 u} d u = \frac{1}{8} ln ∣ u ∣

\int \frac{1}{8 ( u + 8 )} d u = \frac{1}{8} ln ∣ u + 8 ∣

= 9 \cdot \frac{1}{4} (\frac{1}{8} ln ∣ u ∣ - \frac{1}{8} ln ∣ u + 8 ∣)

u = x^{4}

代回

= 9 \cdot \frac{1}{4} (\frac{1}{8} ln x^{4} - \frac{1}{8} ln x^{4} + 8)

化简

9 \cdot \frac{1}{4} (\frac{1}{8} ln x^{4} - \frac{1}{8} ln x^{4} + 8) : \frac{9}{4} (\frac{1}{2} ln (x) - \frac{1}{8} ln x^{4} + 8)

= \frac{9}{4} (\frac{1}{2} ln (x) - \frac{1}{8} ln x^{4} + 8)

解答补常数

= \frac{9}{4} (\frac{1}{2} ln (x) - \frac{1}{8} ln x^{4} + 8) + C

作图

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