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

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

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

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

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

Solution steps

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

Take the constant out:

\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

Expand

\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

Apply the Sum Rule:

\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 ∣)

Add a constant to the solution

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

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