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

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

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

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

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

Solution steps

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

Simplify

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

Expand

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

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

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

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

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

Add a constant to the solution

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

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