What is the integral from 1 to e of (ln(x))^8 ?

The integral from 1 to e of (ln(x))^8 is 14833e-40320

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integral from 1 to e of (ln(x))^8

\int_{1}^{e} (ln (x))^{8} d x

14833 e - 40320

Decimal Notation

0.27436 \dots

Solution steps

\int_{1}^{e} (ln (x))^{8} d x

Simplify

= \int_{1}^{e} ln^{8} (x) d x

Apply u-substitution

: \int_{0}^{1} u^{8} e^{u} d u

= \int_{0}^{1} u^{8} e^{u} d u

Apply Integration By Parts

: [u^{8} e^{u} - \int 8 u^{7} e^{u} d u]_{0}^{1}

= [u^{8} e^{u} - \int 8 u^{7} e^{u} d u]_{0}^{1}

\int 8 u^{7} e^{u} d u = 8 (e^{u} u^{7} - 7 (e^{u} u^{6} - 6 (e^{u} u^{5} - 5 (e^{u} u^{4} - 4 (e^{u} u^{3} - 3 (e^{u} u^{2} - 2 (e^{u} u - e^{u})))))))

= [u^{8} e^{u} - 8 (e^{u} u^{7} - 7 (e^{u} u^{6} - 6 (e^{u} u^{5} - 5 (e^{u} u^{4} - 4 (e^{u} u^{3} - 3 (e^{u} u^{2} - 2 (e^{u} u - e^{u})))))))]_{0}^{1}

Compute the boundaries:

14833 e - 40320

= 14833 e - 40320

\int_{0}^{π} cos (\frac{5 x}{2}) d x

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What is the integral from 1 to e of (ln(x))^8 ?
The integral from 1 to e of (ln(x))^8 is 14833e-40320