What is the integral of x/(ln(x^3)) ?

The integral of x/(ln(x^3)) is 1/3 \Ei(2ln(x))+C

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integral of x/(ln(x^3))

\int \frac{x}{ln ( x ^{3} )} d x

\frac{1}{3} Ei (2 ln (x)) + C

Solution steps

\int \frac{x}{ln ( x ^{3} )} d x

Simplify

ln (x^{3}) : 3 ln (x)

= \int \frac{x}{3 ln ( x )} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int \frac{x}{ln ( x )} d x

Apply u-substitution

= \frac{1}{3} \cdot \int \frac{e ^{2 u}}{u} d u

Apply u-substitution

= \frac{1}{3} \cdot \int \frac{e ^{v}}{v} d v

Use the nonelementary integral:

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

= \frac{1}{3} Ei (v)

Substitute back

= \frac{1}{3} Ei (2 ln (x))

Add a constant to the solution

= \frac{1}{3} Ei (2 ln (x)) + C

What is the integral of x/(ln(x^3)) ?
The integral of x/(ln(x^3)) is 1/3 \Ei(2ln(x))+C