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

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

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

integral

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

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

Solution steps

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

Apply exponent rule:

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

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

Apply Integration By Parts

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

= - \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

\frac{x ^{3}}{( 1 - x ^{4} )}

sin^{2} (\frac{x}{2})

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

y^{- 1}

f (x) = (x^{2} + 2 x + 5)^{\frac{1}{2}}

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