use by parts to get the result[1]
Let I_n=\int \frac{dx}{(a+\sin{x})^n}
Find the relations between I_n,I_{n+1}\ \text{and}\ I_{n+2}
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Let I_n=\int \frac{dx}{(a+\sin{x})^n}
Find the relations between I_n,I_{n+1}\ \text{and}\ I_{n+2}