What is the integral from 1 to 5 of ((ln(y))/(xy)) ?

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

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integral from 1 to 5 of ((ln(y))/(xy))

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

= \frac{1}{x} \cdot \int_{0}^{l n (5)} u d u

Apply the Power Rule

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

= \frac{1}{x} [\frac{u ^{2}}{2}]_{0}^{l n (5)}

Compute the boundaries:

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

= \frac{1}{x} \cdot \frac{ln ^{2} ( 5 )}{2}

Simplify

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

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