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

The integral of ((2ln(x))^2)/(4x) is 1/3 ln^3(x)+C

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

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

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

Solution steps

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

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{( 2 ln ( x ) ) ^{2}}{x} d x

Apply Integration By Parts

= \frac{1}{4} (4 ln^{2} (x) - \int \frac{8 ln ( x ) - 4 ln ^{2} ( x )}{x} d x)

\int \frac{8 ln ( x ) - 4 ln ^{2} ( x )}{x} d x = 4 ln^{2} (x) - \frac{4}{3} ln^{3} (x)

= \frac{1}{4} (4 ln^{2} (x) - (4 ln^{2} (x) - \frac{4}{3} ln^{3} (x)))

Simplify

\frac{1}{4} (4 ln^{2} (x) - (4 ln^{2} (x) - \frac{4}{3} ln^{3} (x))) : \frac{1}{3} ln^{3} (x)

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

Add a constant to the solution

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

What is the integral of ((2ln(x))^2)/(4x) ?
The integral of ((2ln(x))^2)/(4x) is 1/3 ln^3(x)+C