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

The integral from 1 to 5 of (ln(x))/(x^3) is 6/25-1/50 ln(5)

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

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

\frac{6}{25} - \frac{1}{50} ln (5)

Decimal

0.20781 \dots

Solution steps

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

Apply Integration By Parts

: [- \frac{ln ( x )}{2 x ^{2}} - \int - \frac{1}{2 x ^{3}} d x]_{1}^{5}

= [- \frac{ln ( x )}{2 x ^{2}} - \int - \frac{1}{2 x ^{3}} d x]_{1}^{5}

\int - \frac{1}{2 x ^{3}} d x = \frac{1}{4 x ^{2}}

= [- \frac{ln ( x )}{2 x ^{2}} - \frac{1}{4 x ^{2}}]_{1}^{5}

Compute the boundaries:

\frac{6}{25} - \frac{1}{50} ln (5)

= \frac{6}{25} - \frac{1}{50} ln (5)

\int x + 0.5 d x

\frac{1}{1 - x}, - 1

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\int \frac{2 arctan ( x )}{1 + x ^{2}} d x

What is the integral from 1 to 5 of (ln(x))/(x^3) ?
The integral from 1 to 5 of (ln(x))/(x^3) is 6/25-1/50 ln(5)