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

Take the constant out:

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

= - \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