What is the integral from 0 to 1 of x^2ln((1/x)^3) ?

The integral from 0 to 1 of x^2ln((1/x)^3) is 1/3

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integral from 0 to 1 of x^2ln((1/x)^3)

\int_{0}^{1} x^{2} ln ((\frac{1}{x})^{3}) d x

\frac{1}{3}

Decimal Notation

0.33333 \dots

Solution steps

\int_{0}^{1} x^{2} ln ((\frac{1}{x})^{3}) d x

Simplify

x^{2} ln ((\frac{1}{x})^{3}) : - x^{2} ln (x^{3})

= \int_{0}^{1} - x^{2} ln (x^{3}) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int_{0}^{1} x^{2} ln (x^{3}) d x

Apply Integration By Parts

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

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

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

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

\frac{1}{3} x^{3} ln (x^{3}) = x^{3} ln (x)

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

Compute the boundaries:

- \frac{1}{3}

= - (- \frac{1}{3})

Simplify

= \frac{1}{3}

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What is the integral from 0 to 1 of x^2ln((1/x)^3) ?
The integral from 0 to 1 of x^2ln((1/x)^3) is 1/3