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

The integral from e to e^3 of (4(ln(x))^3)/x is 80

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

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

80

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int_{1}^{3} u^{3} d u

= 4 \cdot \int_{1}^{3} u^{3} d u

Apply the Power Rule

: [\frac{u ^{4}}{4}]_{1}^{3}

= 4 [\frac{u ^{4}}{4}]_{1}^{3}

Compute the boundaries:

20

= 4 \cdot 20

Simplify

= 80

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What is the integral from e to e^3 of (4(ln(x))^3)/x ?
The integral from e to e^3 of (4(ln(x))^3)/x is 80