What is the integral of ((log_{e}(x))^2)/x ?

The integral of ((log_{e}(x))^2)/x is (ln^3(x))/3+C

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integral of ((log_{e}(x))^2)/x

\int \frac{( lo g _{e} ( x ) ) ^{2}}{x} d x

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

Solution steps

\int \frac{( lo g _{e} ( x ) ) ^{2}}{x} d x

Simplify

lo g_{e} (x) : ln (x)

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

Apply u-substitution

: \int u^{2} d u

= \int u^{2} d u

Apply the Power Rule

: \frac{u ^{3}}{3}

= \frac{u ^{3}}{3}

Substitute back

u = ln (x)

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

Add a constant to the solution

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

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What is the integral of ((log_{e}(x))^2)/x ?
The integral of ((log_{e}(x))^2)/x is (ln^3(x))/3+C