Q:

Reduction formula of primitive $\big(1-\sin^3{x}\big)^n\cos{x}$

I am trying to obtain a reduction formula for $$\int_0^{\pi/2}\big(1-\sin^3{x}\big)^n\cos{x}\;\mathrm dx $$ where $n \in \mathbb{N}$. My attempt is as follows $$\text{let } v = \sin{x}\; \implies \;\mathrm dv = (\cos{x})\;\mathrm dx $$ The integral then becomes $$ \int_0^1 (1-v^3)^n\;\mathrm dv $$ By parts$$ 3n\int_0^1 v^3(1-v^3)^{n-1}\;\mathrm dv $$ But I can't get it in the form originally to have the integral exactly the same but in terms of $n-1$. Thanks in advance. 

A:

Continuing from where you left:
$$I_n=3n\int_0^1 v^3(1-v^3)^{n-1}\,dv=3n\left(\int_0^1 (1-v^3)^{n-1}\,dv-\int_0^1(1-v^3)^{n}\,dv\right)$$
$$\Rightarrow I_n=3n\left(I_{n-1}-I_n\right) \Rightarrow I_n=\frac{3n}{3n+1}I_{n-1}$$

