What is the integral from 1 to 3 of (ln(x))/(x^5) ?

The integral from 1 to 3 of (ln(x))/(x^5) is 5/81-1/324 ln(3)

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integral from 1 to 3 of (ln(x))/(x^5)

\int_{1}^{3} \frac{ln ( x )}{x ^{5}} d x

\frac{5}{81} - \frac{1}{324} ln (3)

Decimal Notation

0.05833 \dots

Solution steps

\int_{1}^{3} \frac{ln ( x )}{x ^{5}} d x

Apply Integration By Parts

: [- \frac{ln ( x )}{4 x ^{4}} - \int - \frac{1}{4 x ^{5}} d x]_{1}^{3}

= [- \frac{ln ( x )}{4 x ^{4}} - \int - \frac{1}{4 x ^{5}} d x]_{1}^{3}

\int - \frac{1}{4 x ^{5}} d x = \frac{1}{16 x ^{4}}

= [- \frac{ln ( x )}{4 x ^{4}} - \frac{1}{16 x ^{4}}]_{1}^{3}

Compute the boundaries:

\frac{5}{81} - \frac{1}{324} ln (3)

= \frac{5}{81} - \frac{1}{324} ln (3)

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What is the integral from 1 to 3 of (ln(x))/(x^5) ?
The integral from 1 to 3 of (ln(x))/(x^5) is 5/81-1/324 ln(3)