What is the integral of (4(ln(x))^2)/(x^2) ?

The integral of (4(ln(x))^2)/(x^2) is 4(-(ln^2(x))/x+2(-(ln(x))/x-1/x))+C

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integral of (4(ln(x))^2)/(x^2)

\int \frac{4 ( ln ( x ) ) ^{2}}{x ^{2}} d x

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

Solution steps

\int \frac{4 ( ln ( x ) ) ^{2}}{x ^{2}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \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

= 4 (- \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})

= 4 (- \frac{ln ^{2} ( x )}{x} - (- 2 (- \frac{ln ( x )}{x} - \frac{1}{x})))

Simplify

= 4 (- \frac{ln ^{2} ( x )}{x} + 2 (- \frac{ln ( x )}{x} - \frac{1}{x}))

Add a constant to the solution

= 4 (- \frac{ln ^{2} ( x )}{x} + 2 (- \frac{ln ( x )}{x} - \frac{1}{x})) + C

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What is the integral of (4(ln(x))^2)/(x^2) ?
The integral of (4(ln(x))^2)/(x^2) is 4(-(ln^2(x))/x+2(-(ln(x))/x-1/x))+C