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

The integral of (1+ln(x))/(xln(x^2)) is 1/2 ln(2|ln(x)|)+1/2 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 (2 ∣ ln (x) ∣) + \frac{1}{2} ln (x) + C

Solution steps

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

Expand

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

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

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

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

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

Add a constant to the solution

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