What is the integral of ((2+ln^2(x)))/(x(1-ln(x))) ?

The integral of ((2+ln^2(x)))/(x(1-ln(x))) is -1/2 ln^2(x)-3ln|-ln(x)+1|-ln(x)-1/2+2+C

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

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

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

Solution steps

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

Expand

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of ((2+ln^2(x)))/(x(1-ln(x))) ?
The integral of ((2+ln^2(x)))/(x(1-ln(x))) is -1/2 ln^2(x)-3ln|-ln(x)+1|-ln(x)-1/2+2+C