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受欢迎的微积分 >

integral of 5/(x(x^4+9))

解答

\int \frac{5}{x ( x ^{4} + 9 )} d x

解答

\frac{5}{9} ln (x) - \frac{5}{36} ln x^{4} + 9 + C

求解步骤

\int \frac{5}{x ( x ^{4} + 9 )} d x

提出常数:

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

= 5 \cdot \int \frac{1}{x ( x ^{4} + 9 )} d x

使用换元积分法

= 5 \cdot \int \frac{1}{4 u ( u + 9 )} d u

提出常数:

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

= 5 \cdot \frac{1}{4} \cdot \int \frac{1}{u ( u + 9 )} d u

将

\frac{1}{u ( u + 9 )}

用部份分式展开:

\frac{1}{9 u} - \frac{1}{9 ( u + 9 )}

= 5 \cdot \frac{1}{4} \cdot \int \frac{1}{9 u} - \frac{1}{9 ( u + 9 )} 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

= 5 \cdot \frac{1}{4} (\int \frac{1}{9 u} d u - \int \frac{1}{9 ( u + 9 )} d u)

\int \frac{1}{9 u} d u = \frac{1}{9} ln ∣ u ∣

\int \frac{1}{9 ( u + 9 )} d u = \frac{1}{9} ln ∣ u + 9 ∣

= 5 \cdot \frac{1}{4} (\frac{1}{9} ln ∣ u ∣ - \frac{1}{9} ln ∣ u + 9 ∣)

u = x^{4}

代回

= 5 \cdot \frac{1}{4} (\frac{1}{9} ln x^{4} - \frac{1}{9} ln x^{4} + 9)

化简

5 \cdot \frac{1}{4} (\frac{1}{9} ln x^{4} - \frac{1}{9} ln x^{4} + 9) : \frac{5}{9} ln (x) - \frac{5}{36} ln x^{4} + 9

= \frac{5}{9} ln (x) - \frac{5}{36} ln x^{4} + 9

解答补常数

= \frac{5}{9} ln (x) - \frac{5}{36} ln x^{4} + 9 + C

作图

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