How do you evaluate the indefinite integral of sin(t) * cos(t) dt, and what is the constant of integration?

To evaluate the indefinite integral of sin(t) * cos(t) with respect to t, we can approach it by using a substitution method or applying a trigonometric identity.

One effective method is to utilize the identity:

sin(t) * cos(t) = 1/2 * sin(2t)

Using this identity, we can rewrite the integral:

   ∫ sin(t) * cos(t) dt = ∫ (1/2) * sin(2t) dt

Now, we can factor out the constant:

   = 1/2 * ∫ sin(2t) dt

Next, we integrate sin(2t). The integral of sin(kt) is -1/k * cos(kt), where k is a constant. Here k=2, so:

   ∫ sin(2t) dt = -1/2 * cos(2t)

Substituting this back into our equation gives:

   1/2 * ∫ sin(2t) dt = 1/2 * (-1/2 * cos(2t))

This simplifies to:

   = -1/4 * cos(2t)

Finally, we must remember to add the constant of integration, denoted as C, to our final result:

   ∫ sin(t) * cos(t) dt = -1/4 * cos(2t) + C

In summary, the indefinite integral of sin(t) * cos(t) with respect to t is:

   = -1/4 * cos(2t) + C