Integrate the following functions:f(x)=sin⁡(5x)cos⁡(5x)

To integrate the function f(x) = sin(5x)cos(5x), we can use the power-reducing identity, which states that 2sin(A)cos(A) = sin(2A).

Step 1: Rewrite the function using the power-reducing identity f(x) = 1/2 sin(2*5x) = 1/2 sin(10x)

Step 2: Integrate the function ∫f(x) dx = ∫1/2 sin(10x) dx

Step 3: Use the integral of sin(ax), which is -1/a cos(ax) ∫f(x) dx = -1/20 cos(10x) + C

So, the integral of sin(5x)cos(5x) is -1/20 cos(10x) + C, where C is the constant of integration.