What is the integral from 1 to e of (ln(x^3))/x ?

The integral from 1 to e of (ln(x^3))/x is 3/2

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integral from 1 to e of (ln(x^3))/x

\int_{1}^{e} \frac{ln ( x ^{3} )}{x} d x

\frac{3}{2}

Decimal

1.5

Solution steps

\int_{1}^{e} \frac{ln ( x ^{3} )}{x} d x

Apply u-substitution

: \int_{1}^{e^{3}} \frac{ln ( u )}{3 u} d u

= \int_{1}^{e^{3}} \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_{1}^{e^{3}} \frac{ln ( u )}{u} d u

Apply u-substitution

: \int_{0}^{3} v d v

= \frac{1}{3} \cdot \int_{0}^{3} v d v

Apply the Power Rule

: [\frac{v ^{2}}{2}]_{0}^{3}

= \frac{1}{3} [\frac{v ^{2}}{2}]_{0}^{3}

Compute the boundaries:

\frac{9}{2}

= \frac{1}{3} \cdot \frac{9}{2}

Simplify

= \frac{3}{2}

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What is the integral from 1 to e of (ln(x^3))/x ?
The integral from 1 to e of (ln(x^3))/x is 3/2