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

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

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

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

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

Solution steps

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

Simplify

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

Apply u-substitution

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

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

Apply Integration By Parts

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

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

\int - \frac{2 e ^{2 u}}{u} d u = - 2 Ei (2 u)

= - \frac{e ^{2 u}}{u} - (- 2 Ei (2 u))

Substitute back

u = ln (x)

= - \frac{e ^{2 l n (x)}}{ln ( x )} - (- 2 Ei (2 ln (x)))

Simplify

- \frac{e ^{2 l n (x)}}{ln ( x )} - (- 2 Ei (2 ln (x))) : - \frac{x ^{2}}{ln ( x )} + 2 Ei (2 ln (x))

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

Add a constant to the solution

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

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