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

To integrate sin^2(wt) with respect to time, we can use the trigonometric identity that relates sin^2(x) to 1/2 - 1/2cos(2x).

Trigonometric identity:
sin^2(x) = 1/2 - 1/2cos(2x)

Using this identity, we can rewrite sin^2(wt) as:

sin^2(wt) = 1/2 - 1/2cos(2wt)

Now, we can proceed with integrating sin^2(wt) by breaking it down into two separate integrals:

1. Integral of 1/2:
The integral of a constant is equal to that constant multiplied by the variable of integration. Therefore, the integral of 1/2 with respect to time is:

∫(1/2) dt = (1/2)t + C1

2. Integral of -1/2cos(2wt):
To integrate -1/2cos(2wt) with respect to time, we can use the trigonometric identity for the integral of cos(ax) which is sin(ax)/a. In this case, a = 2w.

Using this identity, the integral of -1/2cos(2wt) is:

∫(-1/2cos(2wt)) dt = (-1/2)(sin(2wt)/(2w)) + C2

Where C1 and C2 are constants of integration.

Final integral:
Combining the results of the two integrals, we get the final integral of sin^2(wt) with respect to time as:

∫sin^2(wt) dt = (1/2)t - (1/4w)sin(2wt) + C

Where C = C1 + C2 is the final constant of integration.

Conclusion:
The integral of sin^2(wt) with respect to time is given by (1/2)t - (1/4w)sin(2wt) + C, where C is the constant of integration. This result can be obtained by using the trigonometric identity sin^2(x) = 1/2 - 1/2cos(2x) and integrating each term separately.