What is the integral from 1 to 7 of (ln(x^2))/(x^3) ?

The integral from 1 to 7 of (ln(x^2))/(x^3) is 24/49-1/49 ln(7)

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integral from 1 to 7 of (ln(x^2))/(x^3)

\int_{1}^{7} \frac{ln ( x ^{2} )}{x ^{3}} d x

\frac{24}{49} - \frac{1}{49} ln (7)

Decimal Notation

0.45008 \dots

Solution steps

\int_{1}^{7} \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}^{7}

= [- \frac{ln ( x ^{2} )}{2 x ^{2}} - \int - \frac{1}{x ^{3}} d x]_{1}^{7}

\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}^{7}

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

= [- \frac{ln ( x )}{x ^{2}} - \frac{1}{2 x ^{2}}]_{1}^{7}

Compute the boundaries:

\frac{24}{49} - \frac{1}{49} ln (7)

= \frac{24}{49} - \frac{1}{49} ln (7)

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What is the integral from 1 to 7 of (ln(x^2))/(x^3) ?
The integral from 1 to 7 of (ln(x^2))/(x^3) is 24/49-1/49 ln(7)