# What is int1/(xln^3x)dx ?

## $\int \frac{1}{x {\ln}^{3} x} \mathrm{dx}$

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#### Explanation

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#### Explanation:

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Hammer Share
Apr 20, 2018

$\int \frac{1}{x {\ln}^{3} x} \mathrm{dx} = - \frac{1}{2 {\ln}^{2} x} + C$

#### Explanation:

We will use integration by substitution:

Let $u = \ln x$. Then $\frac{\mathrm{du}}{\mathrm{dx}} = \frac{1}{x} \implies \mathrm{dx} = x \mathrm{du}$.

Thus,

$\int \frac{1}{x {\ln}^{3} x} \mathrm{dx} = \int \frac{1}{x {u}^{3}} \cdot x \mathrm{du} = \int \frac{1}{u} ^ 3 \mathrm{du}$

As you can see, the $x$ cancells nicely.

We have to apply the power rule now:

$\int {u}^{n} \mathrm{du} = {u}^{n + 1} / \left(n + 1\right) + C$ with $n = - 3$.

$\int \frac{1}{u} ^ 3 \mathrm{du} = {u}^{- 2} / - 2 + C = - \frac{1}{2 {u}^{2}} + C$

Undo the substitution :

$= - \frac{1}{2 {\ln}^{2} x} + C$

color(red)( :. int 1/(xln^3x) dx =-1/(2ln^2x) + C

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