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

The integral of (1+ln(x))/((xln(x))^2) is -1/(xln(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}{x ln ( x )} + C

Solution steps

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

(x ln (x))^{2} = x^{2} ln^{2} (x)

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

Expand

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

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

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

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

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

Simplify

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

Add a constant to the solution

= - \frac{1}{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/(xln(x))+C