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

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\frac{\partial}{\partial y} (x cos (x + y))

\frac{d y}{d x} + 2 y = 20 e^{3 x}

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