find an antiderivative P of p(s)=2sin(2s)

d/dx cos u = -sin u du/dx

so

-cos 2s times something

try

-(1/2)cos 2s

P = - cos(2s) + C

To find the antiderivative P of p(s) = 2sin(2s), we can use integration techniques.

Step 1: Recall the integral of sin(x) is -cos(x) + C, where C is the constant of integration.

Step 2: We can use a substitution to simplify the integral. Let u = 2s, and then du = 2 ds, which implies ds = (1/2) du.

Step 3: Substitute u = 2s and ds = (1/2) du into the integral:

∫2sin(2s) ds = ∫2sin(u) * (1/2) du

Step 4: Simplify the integral:

= ∫sin(u) du

Step 5: Apply the integration rule for sin(u):

= -cos(u) + C

Step 6: Substitute back u = 2s:

= -cos(2s) + C

Therefore, the antiderivative P of p(s) = 2sin(2s) is P(s) = -cos(2s) + C, where C is the constant of integration.

To find an antiderivative of the function p(s) = 2sin(2s), you can follow these steps:

Step 1: Identify the function's basic form.
The given function, p(s) = 2sin(2s), is a product of a constant (2) and a trigonometric function (sin(2s)).

Step 2: Recognize the derivatives of basic trigonometric functions.
The derivative of sin(s) is cos(s). However, we have sin(2s) in the given function, so we need to adjust accordingly.

Step 3: Apply the power rule for integration.
To integrate sin(2s), we need to reverse the process of differentiation. The power rule states that ∫ sin(ax) dx = -(1/a)cos(ax) + C, where 'a' is a constant.

Since we have sin(2s), the constant 'a' is 2. Therefore, ∫ sin(2s) ds = -(1/2)cos(2s) + C1, where C1 is the constant of integration.

Step 4: Adjust for the constant multiplier.
Given p(s) = 2sin(2s), we need to multiply the result from Step 3 by the constant '2' to match the original function.

Multiplying -(1/2)cos(2s) by 2, we obtain P(s) = -cos(2s) + C2, where C2 = 2C1 is the new constant of integration.

Thus, the antiderivative of p(s) = 2sin(2s) is P(s) = -cos(2s) + C2, where C2 represents the constant of integration.