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

The integral from 1 to e of ((1+ln(x))^4)/x is 31/5

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

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

\frac{31}{5}

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{4 ( ln ( x ) + 1 ) ^{3} - ( ln ( x ) + 1 ) ^{4}}{x} d x = (ln (x) + 1)^{4} - \frac{( ln ( x ) + 1 ) ^{5}}{5}

= [(1 + ln (x))^{4} - ((ln (x) + 1)^{4} - \frac{( ln ( x ) + 1 ) ^{5}}{5})]_{1}^{e}

Simplify

[(1 + ln (x))^{4} - ((ln (x) + 1)^{4} - \frac{( ln ( x ) + 1 ) ^{5}}{5})]_{1}^{e} : [\frac{1}{5} (ln (x) + 1)^{5}]_{1}^{e}

= [\frac{1}{5} (ln (x) + 1)^{5}]_{1}^{e}

Compute the boundaries:

\frac{31}{5}

= \frac{31}{5}

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What is the integral from 1 to e of ((1+ln(x))^4)/x ?
The integral from 1 to e of ((1+ln(x))^4)/x is 31/5