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

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

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

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

diverges

Solution steps

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

Apply Integration By Parts

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

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

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

= [ln^{3} (x) - (ln^{3} (x) - \frac{ln ^{4} ( x )}{4})]_{0}^{1}

Simplify

[ln^{3} (x) - (ln^{3} (x) - \frac{ln ^{4} ( x )}{4})]_{0}^{1} : [\frac{1}{4} ln^{4} (x)]_{0}^{1}

= [\frac{1}{4} ln^{4} (x)]_{0}^{1}

Compute the boundaries:

diverges

= diverges

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\int_{- \infty}^{\infty} 0 d x

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\int_{0}^{1} [2 x] sin (π x) d x

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