<s>[INST] Integral with un-intuitive U-Substitution On a recent midterm we were given the following integral and were expected to integrate it.
$$\int_{-27}^{-8} \frac{\mathrm dx}{x + x^{2/3}}$$
 [/INST] $$x+x^{\frac{2}{3}}=\\x^{\frac{3}{3}}+x^{\frac{2}{3}}=\\x^{\frac{2}{3}}(x^{\frac{1}{3}}+1)=\\\sqrt[3]{x^2}(\sqrt[3]{x}+1)$$now apply this$$u=\sqrt[3]{x}\\x=u^3\\dx=3u^2du\\\frac{1}{x+x^{\frac{2}{3}}}dx=\\\frac{1}{u^2(u+1)}*(3u^2du)=\\\frac{3du}{u+1}\\$$so $$\int\frac{1}{x+x^{\frac{2}{3}}}dx=\int  \frac{3du}{u+1}=3ln|u+1|+c\\=3ln|\sqrt[3]{x}+1|+c$$
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