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

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

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

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

- \frac{ln ^{3} ( x )}{2 x ^{2}} + \frac{3 ( - 2 ln ^{2} ( x ) - 2 ln ( x ) - 1 )}{8 x ^{2}} + C

Solution steps

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

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

Apply Integration By Parts

= - \frac{ln ^{3} ( x )}{2 x ^{2}} - \int - \frac{3 ln ^{2} ( x )}{2 x ^{3}} d x

\int - \frac{3 ln ^{2} ( x )}{2 x ^{3}} d x = - \frac{3 ( - 2 ln ^{2} ( x ) - 2 ln ( x ) - 1 )}{8 x ^{2}}

= - \frac{ln ^{3} ( x )}{2 x ^{2}} - (- \frac{3 ( - 2 ln ^{2} ( x ) - 2 ln ( x ) - 1 )}{8 x ^{2}})

Simplify

= - \frac{ln ^{3} ( x )}{2 x ^{2}} + \frac{3 ( - 2 ln ^{2} ( x ) - 2 ln ( x ) - 1 )}{8 x ^{2}}

Add a constant to the solution

= - \frac{ln ^{3} ( x )}{2 x ^{2}} + \frac{3 ( - 2 ln ^{2} ( x ) - 2 ln ( x ) - 1 )}{8 x ^{2}} + C

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