What is the integral of ((ln(x))^3)/(x^2) ?

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

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integral of ((ln(x))^3)/(x^2)

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

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

Solution steps

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

Simplify

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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