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

The integral from 1/2 to 1 of (ln(x))/(4x) is -(ln^2(2))/8

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

\int_{\frac{1}{2}}^{1} \frac{ln ( x )}{4 x} d x

- \frac{ln ^{2} ( 2 )}{8}

Decimal Notation

- 0.06005 \dots

Solution steps

\int_{\frac{1}{2}}^{1} \frac{ln ( x )}{4 x} d x

Take the constant out:

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

= \frac{1}{4} \cdot \int_{\frac{1}{2}}^{1} \frac{ln ( x )}{x} d x

Apply u-substitution

: \int_{- l n (2)}^{0} u d u

= \frac{1}{4} \cdot \int_{- l n (2)}^{0} u d u

Apply the Power Rule

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

= \frac{1}{4} [\frac{u ^{2}}{2}]_{- l n (2)}^{0}

Compute the boundaries:

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

= \frac{1}{4} (- \frac{ln ^{2} ( 2 )}{2})

Simplify

= - \frac{ln ^{2} ( 2 )}{8}

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What is the integral from 1/2 to 1 of (ln(x))/(4x) ?
The integral from 1/2 to 1 of (ln(x))/(4x) is -(ln^2(2))/8