Q: Given that $\displaystyle I_n = \displaystyle\int^8_0 x^n (8-x)^{\frac{1}{3}} \, \mathrm{d}x, \ \ n \ge 0$.

(a) Show that $\displaystyle I_n = \frac{24n}{3n+4}I_{n-1}, \ \ n \ge 1$.

(b) Hence find the exact value of $\displaystyle \displaystyle\int^8_0 x(x+5)(8-x)^{\frac{1}{3}} \, \mathrm{d}x$.

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I've done part (a) and got $\displaystyle I_n = 6nI_{n-1} - \frac{3}{4}nI_n$ but cannot simplify to get it into the form that the question is asking for. Also, I cannot do part (b). Can I have help? Thanks in advance.