<s>[INST] Integration by parts or using u-sub? Integrate: $4(x^3+1)^{\frac{1}{3}}(x^5)$
I tried using u-sub but it is getting me absolutely nowhere. I also tried using integration by parts and eventually was able to evaluate the integral but it does not seem to simplify to the answer. Thank you for your help.
 [/INST] Maybe this will help: let $u = x^{3} + 1$.  Then $u - 1 = x^{3}$, and $du = 3x^{2}\, dx$ (i.e., $\frac{1}{3} \, du = x^{2} \, dx$), so we get:
$\int 4(x^{3} + 1)^{\frac{1}{3}}(x^{3}x^{2}) \,dx = \int \frac{4}{3}(u)^{\frac{1}{3}}(u - 1) \,du = \frac{4}{3} \int u^{\frac{4}{3}} - u^{\frac{1}{3}} \,du  = \frac{4}{3}(\frac{3}{7}u^{\frac{7}{3}} - \frac{3}{4}u^{\frac{4}{3}}) + C$
$ = \frac{4}{7}u^{\frac{7}{3}} - u^{\frac{4}{3}} + C$.
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