What is the integral of 1/(ln^3(x)) ?

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

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integral of 1/(ln^3(x))

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

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

Solution steps

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

Apply u-substitution

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

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

Apply Integration By Parts

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

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

\int - \frac{e ^{u}}{2 u ^{2}} d u = - \frac{1}{2} (- \frac{e ^{u}}{u} + Ei (u))

= - \frac{e ^{u}}{2 u ^{2}} - (- \frac{1}{2} (- \frac{e ^{u}}{u} + Ei (u)))

Substitute back

u = ln (x)

= - \frac{e ^{l n (x)}}{2 ln ^{2} ( x )} - (- \frac{1}{2} (- \frac{e ^{l n (x)}}{ln ( x )} + Ei (ln (x))))

Simplify

- \frac{e ^{l n (x)}}{2 ln ^{2} ( x )} - (- \frac{1}{2} (- \frac{e ^{l n (x)}}{ln ( x )} + Ei (ln (x)))) : - \frac{x}{2 ln ^{2} ( x )} + \frac{1}{2} (- \frac{x}{ln ( x )} + Ei (ln (x)))

= - \frac{x}{2 ln ^{2} ( x )} + \frac{1}{2} (- \frac{x}{ln ( x )} + Ei (ln (x)))

Add a constant to the solution

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

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