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

The integral of x/(ln^3(x)) is 4(-(x^2)/(8ln^2(x))+1/2 (-(x^2)/(2ln(x))+\Ei(2ln(x))))+C

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

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

4 (- \frac{x ^{2}}{8 ln ^{2} ( x )} + \frac{1}{2} (- \frac{x ^{2}}{2 ln ( x )} + Ei (2 ln (x)))) + C

Solution steps

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

Apply u-substitution

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

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

Apply u-substitution

: \int \frac{4 e ^{v}}{v ^{3}} d v

= \int \frac{4 e ^{v}}{v ^{3}} d v

Take the constant out:

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

= 4 \cdot \int \frac{e ^{v}}{v ^{3}} d v

Apply Integration By Parts

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

= 4 (- \frac{e ^{v}}{2 v ^{2}} - \int - \frac{e ^{v}}{2 v ^{2}} d v)

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

= 4 (- \frac{e ^{v}}{2 v ^{2}} - (- \frac{1}{2} (- \frac{e ^{v}}{v} + Ei (v))))

Substitute back

= 4 (- \frac{e ^{2 l n (x)}}{2 ( 2 ln ( x ) ) ^{2}} - (- \frac{1}{2} (- \frac{e ^{2 l n (x)}}{2 ln ( x )} + Ei (2 ln (x)))))

Simplify

4 (- \frac{e ^{2 l n (x)}}{2 ( 2 ln ( x ) ) ^{2}} - (- \frac{1}{2} (- \frac{e ^{2 l n (x)}}{2 ln ( x )} + Ei (2 ln (x))))) : 4 (- \frac{x ^{2}}{8 ln ^{2} ( x )} + \frac{1}{2} (- \frac{x ^{2}}{2 ln ( x )} + Ei (2 ln (x))))

= 4 (- \frac{x ^{2}}{8 ln ^{2} ( x )} + \frac{1}{2} (- \frac{x ^{2}}{2 ln ( x )} + Ei (2 ln (x))))

Add a constant to the solution

= 4 (- \frac{x ^{2}}{8 ln ^{2} ( x )} + \frac{1}{2} (- \frac{x ^{2}}{2 ln ( x )} + Ei (2 ln (x)))) + C

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What is the integral of x/(ln^3(x)) ?
The integral of x/(ln^3(x)) is 4(-(x^2)/(8ln^2(x))+1/2 (-(x^2)/(2ln(x))+\Ei(2ln(x))))+C