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

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

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

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

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

Solution steps

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

Expand

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

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

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

Add a constant to the solution

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

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