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

The integral of (ln(x))/(3x) is 1/6 ln^2(x)+C

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

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

\frac{1}{6} ln^{2} (x) + C

Solution steps

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

Apply u-substitution

: \int u d u

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

Apply the Power Rule

: \frac{u ^{2}}{2}

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

Substitute back

u = ln (x)

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

Simplify

\frac{1}{3} \cdot \frac{ln ^{2} ( x )}{2} : \frac{1}{6} ln^{2} (x)

= \frac{1}{6} ln^{2} (x)

Add a constant to the solution

= \frac{1}{6} ln^{2} (x) + C

f (x) = \frac{x}{x ^{2} + 121}

\frac{d}{d x} (x^{3} (3 x + 1))

\frac{d}{d x} (\frac{1}{x ^{2} - x - 6})

x \frac{d y}{d x} = y + x e^{\frac{y}{x}}

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

What is the integral of (ln(x))/(3x) ?
The integral of (ln(x))/(3x) is 1/6 ln^2(x)+C