<s>[INST] Reduction Integration Question ($\int_0^1{\frac{x^n}{\sqrt{x+1}} dx}$) What integration by parts things do I use for a reduction formula for $\int_0^1{\frac{x^n}{\sqrt{x+1}} dx}$?
I have tried many different ways of obtaining it without success.
 [/INST] $$I_n=\int\frac{x^ndx}{\sqrt{x+1}}=\int\frac{x^{n-1}(1+x-1)dx}{\sqrt{x+1}} $$
$$=\int x^{n-1}\sqrt{x+1}dx-\int\frac{x^{n-1}dx}{\sqrt{x+1}}=\int x^{n-1}\sqrt{x+1}dx-I_{n-1}$$
Again, $$\int x^{n-1}\sqrt{x+1}dx= \sqrt{x+1}\frac{x^n}n-\int \frac{x^n}{2n\sqrt{x+1}}dx=\sqrt{x+1}\frac{x^n}n-\frac{I_n}{2n}$$
$$\implies I_n=\sqrt{x+1}\frac{x^n}n-\frac{I_n}{2n}-I_{n-1}$$
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