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

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

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

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

\frac{4}{3} ln^{3} (x) + C

Solution steps

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

Simplify

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

: \int v^{2} d v

= \frac{1}{2} \cdot \int v^{2} d v

Apply the Power Rule

: \frac{v ^{3}}{3}

= \frac{1}{2} \cdot \frac{v ^{3}}{3}

Substitute back

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

Simplify

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

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

Add a constant to the solution

= \frac{4}{3} ln^{3} (x) + C

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