What is the integral from 1 to e of (ln(x))^4 ?

The integral from 1 to e of (ln(x))^4 is 9e-24

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integral from 1 to e of (ln(x))^4

\int_{1}^{e} (ln (x))^{4} d x

9 e - 24

Decimal Notation

0.46453 \dots

Solution steps

\int_{1}^{e} (ln (x))^{4} d x

Simplify

= \int_{1}^{e} ln^{4} (x) d x

Apply Integration By Parts

: [x ln^{4} (x) - \int 4 ln^{3} (x) d x]_{1}^{e}

= [x ln^{4} (x) - \int 4 ln^{3} (x) d x]_{1}^{e}

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

= [x ln^{4} (x) - 4 (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x)))]_{1}^{e}

Compute the boundaries:

9 e - 24

= 9 e - 24

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\int_{3}^{5} \frac{1}{x - 3} d x

\int_{0}^{2} x^{2} + 4 d x

What is the integral from 1 to e of (ln(x))^4 ?
The integral from 1 to e of (ln(x))^4 is 9e-24