sin4𝑥 𝑐𝑜𝑠5𝑥 𝑑𝑥

It seems like you want to integrate the function sin(4x)cos(5x) dx. Here are the steps:

1. Use the product-to-sum identities which states that sin(a)cos(b) = 1/2[sin(a+b) + sin(a-b)]. In this case, a = 4x and b = 5x.

2. Substitute the values into the formula: sin(4x)cos(5x) = 1/2[sin(4x+5x) + sin(4x-5x)] = 1/2[sin(9x) + sin(-x)]

3. Now, integrate the function: ∫sin(4x)cos(5x) dx = ∫1/2[sin(9x) + sin(-x)] dx

4. Break down the integral: = 1/2 ∫sin(9x) dx + 1/2 ∫sin(-x) dx

5. Integrate each part separately: = -1/18 cos(9x) - 1/2 cos(-x) + C

So, the integral of sin(4x)cos(5x) dx is -1/18 cos(9x) - 1/2 cos(-x) + C.