What is the integral from 1 to 8 of (4(ln(x))^3)/x ?

The integral from 1 to 8 of (4(ln(x))^3)/x is 81ln^4(2)

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integral from 1 to 8 of (4(ln(x))^3)/x

\int_{1}^{8} \frac{4 ( ln ( x ) ) ^{3}}{x} d x

81 ln^{4} (2)

Solution steps

\int_{1}^{8} \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_{1}^{8} \frac{( ln ( x ) ) ^{3}}{x} d x

Apply u-substitution

: \int_{0}^{3 l n (2)} u^{3} d u

= 4 \cdot \int_{0}^{3 l n (2)} u^{3} d u

Apply the Power Rule

: [\frac{u ^{4}}{4}]_{0}^{3 l n (2)}

= 4 [\frac{u ^{4}}{4}]_{0}^{3 l n (2)}

Compute the boundaries:

\frac{81 ln ^{4} ( 2 )}{4}

= 4 \cdot \frac{81 ln ^{4} ( 2 )}{4}

Simplify

= 81 ln^{4} (2)

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What is the integral from 1 to 8 of (4(ln(x))^3)/x ?
The integral from 1 to 8 of (4(ln(x))^3)/x is 81ln^4(2)