What is the integral from 1 to 5 of (8(ln(x))^3)/x ?

The integral from 1 to 5 of (8(ln(x))^3)/x is 2ln^4(5)

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integral from 1 to 5 of (8(ln(x))^3)/x

\int_{1}^{5} \frac{8 ( ln ( x ) ) ^{3}}{x} d x

2 ln^{4} (5)

Decimal

13.41920 \dots

Solution steps

\int_{1}^{5} \frac{8 ( ln ( x ) ) ^{3}}{x} d x

Take the constant out:

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

= 8 \cdot \int_{1}^{5} \frac{( ln ( x ) ) ^{3}}{x} d x

Apply u-substitution

: \int_{0}^{l n (5)} u^{3} d u

= 8 \cdot \int_{0}^{l n (5)} u^{3} d u

Apply the Power Rule

: [\frac{u ^{4}}{4}]_{0}^{l n (5)}

= 8 [\frac{u ^{4}}{4}]_{0}^{l n (5)}

Compute the boundaries:

\frac{ln ^{4} ( 5 )}{4}

= 8 \cdot \frac{ln ^{4} ( 5 )}{4}

Simplify

= 2 ln^{4} (5)

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What is the integral from 1 to 5 of (8(ln(x))^3)/x ?
The integral from 1 to 5 of (8(ln(x))^3)/x is 2ln^4(5)