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

解答

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

\frac{32}{5} ln^{5} (2) + 64 ln^{4} (2) + 256 ln^{3} (2) + 512 ln^{2} (2) + 512 ln (2)

十进制

701.93507 \dots

求解步骤

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

使用分布积分法

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

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

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

化简

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

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

带入上下限计算后得:

\frac{1}{5} (2 ln (2) + 4)^{5} - \frac{1024}{5}

= \frac{1}{5} (2 ln (2) + 4)^{5} - \frac{1024}{5}

化简

= \frac{32}{5} ln^{5} (2) + 64 ln^{4} (2) + 256 ln^{3} (2) + 512 ln^{2} (2) + 512 ln (2)

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