What is the integral from 0 to 1 of (ln(x))^3 ?

The integral from 0 to 1 of (ln(x))^3 is -6

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integral from 0 to 1 of (ln(x))^3

\int_{0}^{1} (ln (x))^{3} d x

- 6

Solution steps

\int_{0}^{1} (ln (x))^{3} d x

Apply Integration By Parts

: [x ln^{3} (x) - \int 3 ln^{2} (x) d x]_{0}^{1}

= [x ln^{3} (x) - \int 3 ln^{2} (x) d x]_{0}^{1}

\int 3 ln^{2} (x) d x = 3 (x ln^{2} (x) - 2 (x ln (x) - x))

= [x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x))]_{0}^{1}

Compute the boundaries:

- 6

= - 6

\int_{0}^{5} x^{2} - 3 d x

\int_{1}^{5} 1 + x^{2} d x

\int_{3}^{7} \frac{1}{x} d x

\int_{5}^{x} t^{3} d t

\int_{0}^{\frac{π}{2}} (1 - sin (x))^{2} d x

What is the integral from 0 to 1 of (ln(x))^3 ?
The integral from 0 to 1 of (ln(x))^3 is -6