To solve the integral ∫ sin^4(x) cos^5(x) dx, we can use the power-reducing identity and substitution method. Here are the steps:

Step 1: Let u = sin(x), then du = cos(x) dx.

Step 2: Rewrite the integral in terms of u: ∫ u^4 (1 - u^2)^2 du.

Step 3: Expand the integral: ∫ u^4 - 2u^6 + u^8 du.

Step 4: Now, integrate term by term: [1/5 u^5 - 2/7 u^7 + 1/9 u^9] + C.

Step 5: Substitute u = sin(x) back in: [1/5 sin^5(x) - 2/7 sin^7(x) + 1/9 sin^9(x)] + C.

So, ∫ sin^4(x) cos^5(x) dx = [1/5 sin^5(x) - 2/7 sin^7(x) + 1/9 sin^9(x)] + C.

Question

To solve the integral ∫ sin^4(x) cos^5(x) dx, we can use the power-reducing identity and substitution method. Here are the steps:

Step 1: Let u = sin(x), then du = cos(x) dx.

Step 2: Rewrite the integral in terms of u: ∫ u^4 (1 - u^2)^2 du.

Step 3: Expand the integral: ∫ u^4 - 2u^6 + u^8 du.

Step 4: Now, integrate term by term: [1/5 u^5 - 2/7 u^7 + 1/9 u^9] + C.

Step 5: Substitute u = sin(x) back in: [1/5 sin^5(x) - 2/7 sin^7(x) + 1/9 sin^9(x)] + C.

So, ∫ sin^4(x) cos^5(x) dx = [1/5 sin^5(x) - 2/7 sin^7(x) + 1/9 sin^9(x)] + C.

Knowee AI · Accepted Answer

To solve the integral ∫ sin^4(x) cos^5(x) dx, we can use the power-reducing identity and substitution method. Here are the steps:

Step 1: Let u = sin(x), then du = cos(x) dx.

Step 2: Rewrite the integral in terms of u: ∫ u^4 (1 - u^2)^2 du.

Step 3: Expand the integral: ∫ u^4 - 2u^6 + u^8 du.

Step 4: Now, integrate term by term: [1/5 u^5 - 2/7 u^7 + 1/9 u^9] + C.

Step 5: Substitute u = sin(x) back in: [1/5 sin^5(x) - 2/7 sin^7(x) + 1/9 sin^9(x)] + C.

So, ∫ sin^4(x) cos^5(x) dx = [1/5 sin^5(x) - 2/7 sin^7(x) + 1/9 sin^9(x)] + C.

∫ sin4𝑥 𝑐𝑜𝑠5𝑥 𝑑𝑥