׬ sin 𝑥cos4 𝑥 𝑑𝑥

It seems like you want to integrate the function sin(x)cos^4(x) dx. Here are the steps:

Step 1: Use the power-reducing identity The power-reducing identity is cos^2(x) = (1+cos(2x))/2. We can use this to rewrite cos^4(x) as (cos^2(x))^2 and then apply the identity.

cos^4(x) = [(1+cos(2x))/2]^2 = (1 + 2cos(2x) + cos^2(2x))/4

So the integral becomes 1/4 ∫ sin(x)(1 + 2cos(2x) + cos^2(2x)) dx.

Step 2: Break up the integral We can break this integral up into three separate integrals:

1/4 ∫ sin(x) dx + 1/2 ∫ sin(x)cos(2x) dx + 1/4 ∫ sin(x)cos^2(2x) dx.

Step 3: Solve each integral The first integral is -1/4 cos(x).

For the second integral, we can use the double-angle identity sin(2x) = 2sin(x)cos(x). This gives us the integral of sin^2(x)cos(x), which can be solved using u-substitution (u = cos(x), du = -sin(x) dx) to get -1/4 cos^2(x).

The third integral is a bit more complex. We can use the power-reducing identity again to write cos^2(2x) as (1+cos(4x))/2. This gives us the integral of sin(x)(1+cos(4x))/2, which can be broken up into two integrals: 1/2 ∫ sin(x) dx + 1/2 ∫ sin(x)cos(4x) dx. The first of these is -1/2 cos(x), and the second can be solved using the product-to-sum identities to get -1/8 sin(4x).

Step 4: Add up the solutions Adding up all these solutions gives us the final answer:

-1/4 cos(x) - 1/4 cos^2(x) - 1/2 cos(x) - 1/8 sin(4x) + C,

where C is the constant of integration.