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

The integral from 1 to e of (ln(x))/(2x) is 1/4

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

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

\frac{1}{4}

Decimal

0.25

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int_{0}^{1} u d u

= \frac{1}{2} \cdot \int_{0}^{1} u d u

Apply the Power Rule

: [\frac{u ^{2}}{2}]_{0}^{1}

= \frac{1}{2} [\frac{u ^{2}}{2}]_{0}^{1}

Compute the boundaries:

\frac{1}{2}

= \frac{1}{2} \cdot \frac{1}{2}

Simplify

= \frac{1}{4}

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What is the integral from 1 to e of (ln(x))/(2x) ?
The integral from 1 to e of (ln(x))/(2x) is 1/4