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

The integral from e to e^3 of (ln(x^3))/x is 12

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integral from e to e^3 of (ln(x^3))/x

\int_{e}^{e^{3}} \frac{ln ( x ^{3} )}{x} d x

12

Solution steps

\int_{e}^{e^{3}} \frac{ln ( x ^{3} )}{x} d x

Apply u-substitution

= \int_{e^{3}}^{e^{9}} \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_{e^{3}}^{e^{9}} \frac{ln ( u )}{u} d u

Apply u-substitution

= \frac{1}{3} \cdot \int_{3}^{9} v d v

Apply the Power Rule

= \frac{1}{3} [\frac{v ^{2}}{2}]_{3}^{9}

Compute the boundaries:

36

= \frac{1}{3} \cdot 36

Simplify

= 12

What is the integral from e to e^3 of (ln(x^3))/x ?
The integral from e to e^3 of (ln(x^3))/x is 12