What is the integral from 1 to 5 of (2ln(x))/x ?

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

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integral from 1 to 5 of (2ln(x))/x

\int_{1}^{5} \frac{2 ln ( x )}{x} d x

ln^{2} (5)

Decimal

2.59029 \dots

Solution steps

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

Apply u-substitution

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

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

Apply the Power Rule

: [\frac{u ^{2}}{2}]_{0}^{l n (5)}

= 2 [\frac{u ^{2}}{2}]_{0}^{l n (5)}

Compute the boundaries:

\frac{ln ^{2} ( 5 )}{2}

= 2 \cdot \frac{ln ^{2} ( 5 )}{2}

Simplify

= ln^{2} (5)

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