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

\frac{d}{d x} (3 sin^{2} (x) cos (2 x))

\frac{\partial}{\partial x} ((\frac{x}{x - y})^{\frac{1}{3}})

\frac{\partial}{\partial x} (\frac{x - y}{( x + y ) ^{2}})

\frac{d}{d x} (\frac{x ^{2} - 10 x}{( x - 5 ) ^{2}})

f (x) = \frac{2 x}{x ^{2} + 1}

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