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

The integral of x/(ln(x^2)) is 1/2 \Ei(ln(x^2))+C

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{1}{2 ln ( u )} d u

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

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 ( u )} d u

Apply u-substitution

: \int \frac{e ^{v}}{v} d v

= \frac{1}{2} \cdot \int \frac{e ^{v}}{v} d v

Use the nonelementary integral:

\int \frac{e ^{x}}{x} d x = Ei (x)

= \frac{1}{2} Ei (v)

Substitute back

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

Add a constant to the solution

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

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