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

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

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integral of (ln(x^2))/(3x)

\int \frac{ln ( x ^{2} )}{3 x} d x

\frac{1}{3} ln^{2} (x) + C

Solution steps

\int \frac{ln ( x ^{2} )}{3 x} d x

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 ( x ^{2} )}{x} d x

Apply u-substitution

= \frac{1}{3} \cdot \int \frac{ln ( 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}{3} \cdot \frac{1}{2} \cdot \int \frac{ln ( u )}{u} d u

Apply u-substitution

= \frac{1}{3} \cdot \frac{1}{2} \cdot \int v d v

Apply the Power Rule

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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