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受欢迎的微积分 >

integral of (ln(x))/(x(ln^2(x)-ln(x)+1))

解答

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

解答

\frac{1}{2} ln 4 ln^{2} (x) - 4 ln (x) + 4 + \frac{1}{3} arctan (\frac{1}{3} (2 ln (x) - 1)) + C

求解步骤

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

使用换元积分法

= \int \frac{u}{u ^{2} - u + 1} d u

对

u^{2} - u + 1

配方:

(u - \frac{1}{2})^{2} + \frac{3}{4}

= \int \frac{u}{( u - \frac{1}{2} ) ^{2} + \frac{3}{4}} d u

使用换元积分法

= \int \frac{2 ( 2 v + 1 )}{4 v ^{2} + 3} d v

提出常数:

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

= 2 \cdot \int \frac{2 v + 1}{4 v ^{2} + 3} d v

乘开

\frac{2 v + 1}{4 v ^{2} + 3} : \frac{2 v}{4 v ^{2} + 3} + \frac{1}{4 v ^{2} + 3}

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

= 2 (\int \frac{2 v}{4 v ^{2} + 3} d v + \int \frac{1}{4 v ^{2} + 3} d v)

\int \frac{2 v}{4 v ^{2} + 3} d v = \frac{1}{4} ln 4 v^{2} + 3

\int \frac{1}{4 v ^{2} + 3} d v = \frac{1}{2 3} arctan (\frac{2}{3} v)

= 2 (\frac{1}{4} ln 4 v^{2} + 3 + \frac{1}{2 3} arctan (\frac{2}{3} v))

代回

= 2 (\frac{1}{4} ln 4 (ln (x) - \frac{1}{2})^{2} + 3 + \frac{1}{2 3} arctan (\frac{2}{3} (ln (x) - \frac{1}{2})))

化简

2 (\frac{1}{4} ln 4 (ln (x) - \frac{1}{2})^{2} + 3 + \frac{1}{2 3} arctan (\frac{2}{3} (ln (x) - \frac{1}{2}))) : \frac{1}{2} ln 4 ln^{2} (x) - 4 ln (x) + 4 + \frac{1}{3} arctan (\frac{1}{3} (2 ln (x) - 1))

= \frac{1}{2} ln 4 ln^{2} (x) - 4 ln (x) + 4 + \frac{1}{3} arctan (\frac{1}{3} (2 ln (x) - 1))

解答补常数

= \frac{1}{2} ln 4 ln^{2} (x) - 4 ln (x) + 4 + \frac{1}{3} arctan (\frac{1}{3} (2 ln (x) - 1)) + C

作图

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