Integration of sin(2wt) with respect to time:

To integrate sin(2wt) with respect to time, we can use the basic integration rules and the trigonometric identity for the double angle of sine function. Here is the step-by-step solution:

Step 1: Identify the integral:
We want to find the integral of sin(2wt) with respect to time.

Step 2: Apply the trigonometric identity:
The trigonometric identity for the double angle of sine is sin(2θ) = 2sin(θ)cos(θ). Using this identity, we can rewrite sin(2wt) as 2sin(wt)cos(wt).

Step 3: Split the integral:
Now we can split the integral into two parts: the integral of 2sin(wt) and the integral of cos(wt).

Step 4: Integrate sin(wt):
To integrate sin(wt), we can use the power rule for integration. The power rule states that the integral of x^n with respect to x is (1/(n+1)) * x^(n+1) + C, where C is the constant of integration. Applying this rule, we can integrate sin(wt) as -(1/w) * cos(wt).

Step 5: Integrate cos(wt):
To integrate cos(wt), we can also use the power rule. The integral of cos(wt) is (1/w) * sin(wt).

Step 6: Combine the results:
Now we can combine the results from steps 4 and 5. The integral of 2sin(wt) is -2/w * cos(wt), and the integral of cos(wt) is (1/w) * sin(wt). Therefore, the integral of sin(2wt) with respect to time is:

-2/w * cos(wt) + (1/w) * sin(wt) + C,

where C is the constant of integration.

Summary:
To integrate sin(2wt) with respect to time, we used the trigonometric identity sin(2θ) = 2sin(θ)cos(θ) and split the integral into two parts: the integral of 2sin(wt) and the integral of cos(wt). We then applied the power rule for integration to find the integrals of sin(wt) and cos(wt). Finally, we combined the results to obtain the integral of sin(2wt) as -2/w * cos(wt) + (1/w) * sin(wt) + C, where C is the constant of integration.