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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

Simplify

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

Add a constant to the solution

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

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