integral from 4 to 5 of 1/((x-4)^{1/3)}

解答

\int_{4}^{5} \frac{1}{( x - 4 ) ^{\frac{1}{3}}} d x

\frac{3}{2}

十进制

1.5

求解步骤

\int_{4}^{5} \frac{1}{( x - 4 ) ^{\frac{1}{3}}} d x

使用分布积分法

= [\frac{x}{( x - 4 ) ^{\frac{1}{3}}} - \int - \frac{x}{3 ( x - 4 ) ^{\frac{4}{3}}} d x]_{4}^{5}

\int - \frac{x}{3 ( x - 4 ) ^{\frac{4}{3}}} d x = - \frac{x - 12}{2 ( x - 4 ) ^{\frac{1}{3}}}

= [\frac{x}{( x - 4 ) ^{\frac{1}{3}}} - (- \frac{x - 12}{2 ( x - 4 ) ^{\frac{1}{3}}})]_{4}^{5}

化简

[\frac{x}{( x - 4 ) ^{\frac{1}{3}}} - (- \frac{x - 12}{2 ( x - 4 ) ^{\frac{1}{3}}})]_{4}^{5} : [\frac{3}{2} (x - 4)^{\frac{2}{3}}]_{4}^{5}

= [\frac{3}{2} (x - 4)^{\frac{2}{3}}]_{4}^{5}

带入上下限计算后得:

\frac{3}{2}

= \frac{3}{2}

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