Fórmula de reducción de la primitiva $\big(1-\sin^3{x}\big)^n\cos{x}$

Question

Estoy tratando de obtener una fórmula de reducción de $$\int_0^{\pi/2}\big(1-\sin^3{x}\big)^n\cos{x}\;\mathrm dx $$ 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 $n-1$. Gracias de antemano.

Answer 1

Continuar desde donde lo dejaste: $$I_n=3n\int_0^1 v^3(1-v^3)^{n-1}\,dv=3n\left(\int_0^1 (1-v^3)^{n-1}\,dv-\int_0^1(1-v^3)^{n}\,dv\right)$$ $$\Rightarrow I_n=3n\left(I_{n-1}-I_n\right) \Rightarrow I_n=\frac{3n}{3n+1}I_{n-1}$$