What is the integral from 1 to 4 of (12(ln(x))^5)/x ?

The integral from 1 to 4 of (12(ln(x))^5)/x is 128ln^6(2)

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integral from 1 to 4 of (12(ln(x))^5)/x

\int_{1}^{4} \frac{12 ( ln ( x ) ) ^{5}}{x} d x

128 ln^{6} (2)

Decimal

14.19589 \dots

Solution steps

\int_{1}^{4} \frac{12 ( ln ( x ) ) ^{5}}{x} d x

Take the constant out:

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

= 12 \cdot \int_{1}^{4} \frac{( ln ( x ) ) ^{5}}{x} d x

Apply u-substitution

: \int_{0}^{2 l n (2)} u^{5} d u

= 12 \cdot \int_{0}^{2 l n (2)} u^{5} d u

Apply the Power Rule

: [\frac{u ^{6}}{6}]_{0}^{2 l n (2)}

= 12 [\frac{u ^{6}}{6}]_{0}^{2 l n (2)}

Compute the boundaries:

\frac{32 ln ^{6} ( 2 )}{3}

= 12 \cdot \frac{32 ln ^{6} ( 2 )}{3}

Simplify

= 128 ln^{6} (2)

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What is the integral from 1 to 4 of (12(ln(x))^5)/x ?
The integral from 1 to 4 of (12(ln(x))^5)/x is 128ln^6(2)