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

The integral from 1 to 8 of (3(ln(x))^2)/x is 27ln^3(2)

integral from 1 to 8 of (3(ln(x))^2)/x

\int_{1}^{8} \frac{3 ( ln ( x ) ) ^{2}}{x} d x

27 ln^{3} (2)

Decimal

8.99166 \dots

Solution steps

\int_{1}^{8} \frac{3 ( ln ( x ) ) ^{2}}{x} d x

Take the constant out:

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

= 3 \cdot \int_{1}^{8} \frac{( ln ( x ) ) ^{2}}{x} d x

Apply u-substitution

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

Apply the Power Rule

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

Compute the boundaries:

9 ln^{3} (2)

= 3 \cdot 9 ln^{3} (2)

Simplify

= 27 ln^{3} (2)

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What is the integral from 1 to 8 of (3(ln(x))^2)/x ?
The integral from 1 to 8 of (3(ln(x))^2)/x is 27ln^3(2)