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

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

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

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

- \frac{x ^{3}}{ln ( x )} + 3 Ei (3 ln (x)) + C

Solution steps

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

Apply u-substitution

: \int \frac{e ^{3 u}}{u ^{2}} d u

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

Apply Integration By Parts

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

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

\int - \frac{3 e ^{3 u}}{u} d u = - 3 Ei (3 u)

= - \frac{e ^{3 u}}{u} - (- 3 Ei (3 u))

Substitute back

u = ln (x)

= - \frac{e ^{3 l n (x)}}{ln ( x )} - (- 3 Ei (3 ln (x)))

Simplify

- \frac{e ^{3 l n (x)}}{ln ( x )} - (- 3 Ei (3 ln (x))) : - \frac{x ^{3}}{ln ( x )} + 3 Ei (3 ln (x))

= - \frac{x ^{3}}{ln ( x )} + 3 Ei (3 ln (x))

Add a constant to the solution

= - \frac{x ^{3}}{ln ( x )} + 3 Ei (3 ln (x)) + C

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