What is the integral from 0 to 5 of (ln(x))^2 ?

The integral from 0 to 5 of (ln(x))^2 is 5ln^2(5)-2(5ln(5)-5)

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integral from 0 to 5 of (ln(x))^2

\int_{0}^{5} (ln (x))^{2} d x

5 ln^{2} (5) - 2 (5 ln (5) - 5)

Decimal

6.85707 \dots

Solution steps

\int_{0}^{5} (ln (x))^{2} d x

Apply Integration By Parts

: [x ln^{2} (x) - \int 2 ln (x) d x]_{0}^{5}

= [x ln^{2} (x) - \int 2 ln (x) d x]_{0}^{5}

\int 2 ln (x) d x = 2 (x ln (x) - x)

= [x ln^{2} (x) - 2 (x ln (x) - x)]_{0}^{5}

Compute the boundaries:

5 ln^{2} (5) - 2 (5 ln (5) - 5)

= 5 ln^{2} (5) - 2 (5 ln (5) - 5)

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What is the integral from 0 to 5 of (ln(x))^2 ?
The integral from 0 to 5 of (ln(x))^2 is 5ln^2(5)-2(5ln(5)-5)