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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{1}{u ^{3}} d u

= - \int \frac{1}{u ^{3}} d u

Apply the Power Rule

: - \frac{1}{2 u ^{2}}

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

Substitute back

u = ln (x)

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

Simplify

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

Add a constant to the solution

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

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