What is the integral from 1 to 2 of (ln(x^2))/(x^3) ?

The integral from 1 to 2 of (ln(x^2))/(x^3) is 3/8-1/4 ln(2)

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integral from 1 to 2 of (ln(x^2))/(x^3)

\int_{1}^{2} \frac{ln ( x ^{2} )}{x ^{3}} d x

\frac{3}{8} - \frac{1}{4} ln (2)

Decimal

0.20171 \dots

Solution steps

\int_{1}^{2} \frac{ln ( x ^{2} )}{x ^{3}} d x

Apply Integration By Parts

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

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

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

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

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

Compute the boundaries:

\frac{3}{8} - \frac{1}{4} ln (2)

= \frac{3}{8} - \frac{1}{4} ln (2)

What is the integral from 1 to 2 of (ln(x^2))/(x^3) ?
The integral from 1 to 2 of (ln(x^2))/(x^3) is 3/8-1/4 ln(2)