What is the integral from y to e of (ln(x))/x ?

The integral from y to e of (ln(x))/x is (1-ln^2(y))/2

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integral from y to e of (ln(x))/x

\int_{y}^{e} \frac{ln ( x )}{x} d x

\frac{1 - ln ^{2} ( y )}{2}

Solution steps

\int_{y}^{e} \frac{ln ( x )}{x} d x

Apply u-substitution

: \int_{l n (y)}^{1} u d u

= \int_{l n (y)}^{1} u d u

Apply the Power Rule

: [\frac{u ^{2}}{2}]_{l n (y)}^{1}

= [\frac{u ^{2}}{2}]_{l n (y)}^{1}

Compute the boundaries:

\frac{1}{2} - \frac{ln ^{2} ( y )}{2}

= \frac{1}{2} - \frac{ln ^{2} ( y )}{2}

Simplify

= \frac{1 - ln ^{2} ( y )}{2}

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