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Question Number 78251 by aliesam last updated on 15/Jan/20

$$\int \frac{x+4}{x-\sqrt[3]{x}}dx$$

Answered by john santu last updated on 15/Jan/20

$$letx=u^3\Rightarrow dx=3u^{2}du$$$$\int \frac{u^3+4}{u^3-u}\times 3u^{2}du=$$$$\int \frac{3u^4+12u}{u^2-1}du=\int 3u^2+3du+$$$$\int \frac{12u+4}{u^2-1}du=u^3+3u+\int \frac{4}{u-1}+\frac{8}{u+1}du$$$$=u^3+3u+4ln(u-1)+8ln(u+1)+c$$$$=x+3\sqrt[3]{x}+4ln(\sqrt[3]{x}-1)+8ln(\sqrt[3]{x}+1)+c$$

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