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

The integral from 1 to 8 of (2ln(x))/x is 9ln^2(2)

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

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

9 ln^{2} (2)

Decimal

4.32407 \dots

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

Apply the Power Rule

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

Compute the boundaries:

\frac{9 ln ^{2} ( 2 )}{2}

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

Simplify

= 9 ln^{2} (2)

What is the integral from 1 to 8 of (2ln(x))/x ?
The integral from 1 to 8 of (2ln(x))/x is 9ln^2(2)