By using the reduction formula, evaluate the following integral ∫ sin4𝑥 𝑐𝑜𝑠5𝑥 𝑑𝑥

To solve the integral ∫ sin^4(x) cos^5(x) dx using the reduction formula, we can follow these steps:

Step 1: We can rewrite the integral as ∫ sin^4(x) cos^4(x) cos(x) dx. This allows us to use the power-reducing formula.

Step 2: We can then use the substitution method. Let u = sin(x), then du = cos(x) dx.

Step 3: Substitute u and du into the integral, we get ∫ u^4 (1 - u^2)^2 du.

Step 4: Expand the integral to ∫ u^4 - 2u^6 + u^8 du.

Step 5: Integrate term by term to get 1/5 u^5 - 2/7 u^7 + 1/9 u^9 + C.

Step 6: Substitute u = sin(x) back into the integral to get the final answer: 1/5 sin^5(x) - 2/7 sin^7(x) + 1/9 sin^9(x) + C.