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受欢迎的微积分 >

积分 8/((x^3-16x))

解答

积分

\frac{8}{( x ^{3} - 16 x )}

解答

- 8 (\frac{1}{16} ln ∣ x ∣ - \frac{1}{32} ln ∣ x + 4 ∣ - \frac{1}{32} ln ∣ x - 4 ∣) + C

求解步骤

\int \frac{8}{x ^{3} - 16 x} d x

提出常数:

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

= 8 \cdot \int \frac{1}{x ^{3} - 16 x} d x

因式分解

x^{3} - 16 x : - (- x^{3} + 16 x)

= 8 \cdot \int \frac{1}{- ( - x ^{3} + 16 x )} d x

提出常数:

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

= 8 (- \int \frac{1}{- x ^{3} + 16 x} d x)

将

\frac{1}{- x ^{3} + 16 x}

用部份分式展开:

\frac{1}{16 x} - \frac{1}{32 ( x + 4 )} - \frac{1}{32 ( x - 4 )}

= 8 (- \int \frac{1}{16 x} - \frac{1}{32 ( x + 4 )} - \frac{1}{32 ( x - 4 )} d x)

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

= 8 (- (\int \frac{1}{16 x} d x - \int \frac{1}{32 ( x + 4 )} d x - \int \frac{1}{32 ( x - 4 )} d x))

\int \frac{1}{16 x} d x = \frac{1}{16} ln ∣ x ∣

\int \frac{1}{32 ( x + 4 )} d x = \frac{1}{32} ln ∣ x + 4 ∣

\int \frac{1}{32 ( x - 4 )} d x = \frac{1}{32} ln ∣ x - 4 ∣

= 8 (- (\frac{1}{16} ln ∣ x ∣ - \frac{1}{32} ln ∣ x + 4 ∣ - \frac{1}{32} ln ∣ x - 4 ∣))

化简

= - 8 (\frac{1}{16} ln ∣ x ∣ - \frac{1}{32} ln ∣ x + 4 ∣ - \frac{1}{32} ln ∣ x - 4 ∣)

解答补常数

= - 8 (\frac{1}{16} ln ∣ x ∣ - \frac{1}{32} ln ∣ x + 4 ∣ - \frac{1}{32} ln ∣ x - 4 ∣) + C

作图

流行的例子

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in t e g r a l \frac{x}{( x + 1 )}

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in t e g r a l \frac{x}{( x ^{2} + y ^{2} )}

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in t e g r a l \frac{1}{( x ^{2} - a ^{2} )}

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in t e g r a l ln (x^{2} + 1)