What is the integral of (ln(x^3))/x ?

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

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integral of (ln(x^3))/x

\int \frac{ln ( x ^{3} )}{x} d x

\frac{3}{2} ln^{2} (x) + C

Solution steps

\int \frac{ln ( x ^{3} )}{x} d x

Apply u-substitution

: \int \frac{ln ( u )}{3 u} d u

= \int \frac{ln ( u )}{3 u} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{ln ( u )}{u} d u

Apply u-substitution

: \int v d v

= \frac{1}{3} \cdot \int v d v

Apply the Power Rule

: \frac{v ^{2}}{2}

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

= \frac{3}{2} ln^{2} (x) + C

x \to 2 lim (\frac{2 - ∣ x ∣}{2 + x})

\int_{- \frac{π}{3}}^{0} 3 sec^{3} (x) d x

x \to 7 - lim (\frac{4 - 3 x}{x - 7})

\int (x + 5)^{2} d x

\frac{d}{d x} (arctan (\frac{x}{1 - x ^{2}}))

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