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

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

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

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

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

Solution steps

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

Simplify

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

Apply Integration By Parts

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

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

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

= 2 x ln^{2} (x) - 4 (2 x ln (x) - 4 x)

Add a constant to the solution

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

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