What is the integral of (1-ln(x))/(xln(x)) ?

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

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

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

ln ∣ ln (x) ∣ - ln ∣ x ∣ + C

Solution steps

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

Expand

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

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

= ln ∣ ln (x) ∣ - ln ∣ x ∣

Add a constant to the solution

= ln ∣ ln (x) ∣ - ln ∣ x ∣ + C

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