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Question Number 122799 by aurpeyz last updated on 19/Nov/20

Answered by liberty last updated on 19/Nov/20

$$lettingu=1+3x^{-1}\Rightarrow du=-3x^{-2}dx$$$$L\left(x\right)=-\frac{1}{3}\int u^{\frac{2}{3}}du=-\frac{1}{3}\times \frac{3}{5}{u}^{\frac{5}{3}}+c$$$$thus\underset{1}{\stackrel{3}{\int }}{x}^{-2}{(1+3x^{-1})}^{\frac{2}{3}}dx=-\frac{1}{5}{\left[{(1+3x^{-1})}^{\frac{5}{3}}\right]}_1^3$$$$=-\frac{1}{5}[2\sqrt[3]{4}-4\sqrt[3]{16}]=\frac{2}{5}[2\sqrt[3]{16}-\sqrt[3]{4}]$$$$=\frac{2}{5}[4\sqrt[3]{2}-\sqrt[3]{4}]$$

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