What is the integral from 1 to e^3 of (4ln(x))/x ?

The integral from 1 to e^3 of (4ln(x))/x is 18

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integral from 1 to e^3 of (4ln(x))/x

\int_{1}^{e^{3}} \frac{4 ln ( x )}{x} d x

18

Solution steps

\int_{1}^{e^{3}} \frac{4 ln ( x )}{x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int_{1}^{e^{3}} \frac{ln ( x )}{x} d x

Apply u-substitution

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

= 4 \cdot \int_{0}^{3} u d u

Apply the Power Rule

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

= 4 [\frac{u ^{2}}{2}]_{0}^{3}

Compute the boundaries:

\frac{9}{2}

= 4 \cdot \frac{9}{2}

Simplify

= 18

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What is the integral from 1 to e^3 of (4ln(x))/x ?
The integral from 1 to e^3 of (4ln(x))/x is 18