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

The integral from 1 to 2 of (ln^2(x))/(x^3) is 3/16+(-ln^2(2)-ln(2))/8

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

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

\frac{3}{16} + \frac{- ln ^{2} ( 2 ) - ln ( 2 )}{8}

Decimal

0.04079 \dots

Solution steps

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

Apply Integration By Parts

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

\int - \frac{ln ( x )}{x ^{3}} d x = \frac{ln ( x )}{2 x ^{2}} + \frac{1}{4 x ^{2}}

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

Simplify

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

Compute the boundaries:

\frac{3}{16} + \frac{- ln ^{2} ( 2 ) - ln ( 2 )}{8}

= \frac{3}{16} + \frac{- ln ^{2} ( 2 ) - ln ( 2 )}{8}

What is the integral from 1 to 2 of (ln^2(x))/(x^3) ?
The integral from 1 to 2 of (ln^2(x))/(x^3) is 3/16+(-ln^2(2)-ln(2))/8