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受欢迎的微积分 >

integral of 1/(2x(x-1))

解答

\int \frac{1}{2 x ( x - 1 )} d x

解答

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

求解步骤

\int \frac{1}{2 x ( x - 1 )} d x

提出常数:

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

= \frac{1}{2} \cdot \int \frac{1}{x ( x - 1 )} d x

将

\frac{1}{x ( x - 1 )}

用部份分式展开:

- \frac{1}{x} + \frac{1}{x - 1}

= \frac{1}{2} \cdot \int - \frac{1}{x} + \frac{1}{x - 1} 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

= \frac{1}{2} (- \int \frac{1}{x} d x + \int \frac{1}{x - 1} d x)

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

解答补常数

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

作图

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