integral of (1+ln(x))/((xln(x))2)

解答

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

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

求解步骤

\int \frac{1 + ln ( x )}{x ln ( x ) \cdot 2} 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 + ln ( x )}{x ln ( x )} d x

乘开

\frac{1 + ln ( x )}{x ln ( x )} : \frac{1}{x ln ( x )} + \frac{1}{x}

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

解答补常数

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

\int \frac{e ^{3 x}}{1 - e ^{3 x}} d x

\int \frac{x ^{2} + 1}{x ^{2} - 4} d x

\int 400 e^{- 0.1 x} + 8 d x

\int x^{4} e (1 - x^{5}) d x

\int \frac{2}{4 - 9 x ^{2}} d x