What is the integral from 3 to 4 of 1/x log_{e}(x) ?

The integral from 3 to 4 of 1/x log_{e}(x) is 2ln^2(2)-(ln^2(3))/2

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integral from 3 to 4 of 1/x log_{e}(x)

\int_{3}^{4} \frac{1}{x} lo g_{e} (x) d x

2 ln^{2} (2) - \frac{ln ^{2} ( 3 )}{2}

Decimal

0.35743 \dots

Solution steps

\int_{3}^{4} \frac{1}{x} lo g_{e} (x) d x

Simplify

\frac{1}{x} lo g_{e} (x) : \frac{ln ( x )}{x}

= \int_{3}^{4} \frac{ln ( x )}{x} d x

Apply u-substitution

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

Apply the Power Rule

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

Compute the boundaries:

2 ln^{2} (2) - \frac{ln ^{2} ( 3 )}{2}

= 2 ln^{2} (2) - \frac{ln ^{2} ( 3 )}{2}

What is the integral from 3 to 4 of 1/x log_{e}(x) ?
The integral from 3 to 4 of 1/x log_{e}(x) is 2ln^2(2)-(ln^2(3))/2