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

解答

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

- ln (∣ x ∣ ∣ 1 - ln (x) ∣) + C

求解步骤

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

使用换元积分法

= \int - \frac{1}{u} d u

提出常数:

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

= - \int \frac{1}{u} d u

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - ln ∣ u ∣

u = x (1 - ln (x))

代回

= - ln ∣ x (1 - ln (x)) ∣

化简

- ln ∣ x (1 - ln (x)) ∣ : - ln (∣ x ∣ ∣ 1 - ln (x) ∣)

= - ln (∣ x ∣ ∣ 1 - ln (x) ∣)

解答补常数

= - ln (∣ x ∣ ∣ 1 - ln (x) ∣) + C

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