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

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

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

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

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

Solution steps

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

Apply exponent rule:

a^{n} \cdot b^{n} = (a \cdot b)^{n}

x^{3} (ln (x) - 1)^{3} = (x (ln (x) - 1))^{3}

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

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

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

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

Add a constant to the solution

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

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