What is the integral of (3x^2)/(ln(x)) ?

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

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integral of (3x^2)/(ln(x))

\int \frac{3 x ^{2}}{ln ( x )} d x

3 Ei (3 ln (x)) + C

Solution steps

\int \frac{3 x ^{2}}{ln ( x )} d x

Take the constant out:

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

= 3 \cdot \int \frac{x ^{2}}{ln ( x )} d x

Apply u-substitution

: \int \frac{e ^{3 u}}{u} d u

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

Apply u-substitution

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

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

Use the nonelementary integral:

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

= 3 Ei (v)

Substitute back

= 3 Ei (3 ln (x))

Add a constant to the solution

= 3 Ei (3 ln (x)) + C

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