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}

(e^{- 4 x})^{^{'}}

\int 30 x^{2} d x

x \to 9 lim (x - 9)

x \to + 8 lim (x^{(\frac{2}{3})})

\frac{d}{d x} (\frac{8 x ^{2} - 2 x + 1}{x ^{2} - 5 x})

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)