Find the average value fave of the function f on the given interval.f(t) = esin(t) cos(t),    0, 𝜋2

The average value of a function f(x) on the interval [a, b] is given by the formula:

fave = 1/(b - a) ∫ from a to b f(x) dx

Here, our function f(t) = e^t * sin(t) * cos(t) and the interval is [0, 𝜋/2].

So, we need to compute the integral of f(t) from 0 to 𝜋/2 and divide by the length of the interval (𝜋/2 - 0 = 𝜋/2).

The integral ∫ from 0 to 𝜋/2 e^t * sin(t) * cos(t) dt is a bit tricky. We can use the product-to-sum identities to simplify the integral.

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

So, our integral becomes ∫ from 0 to 𝜋/2 e^t * 1/2 * sin(2t) dt.

Now, we can use integration by parts, where u = e^t, dv = 1/2 * sin(2t) dt. Then, du = e^t dt, and v = -1/4 * cos(2t).

Using the formula for integration by parts ∫ u dv = uv - ∫ v du, we get:

∫ from 0 to 𝜋/2 e^t * 1/2 * sin(2t) dt = [ -1/4 * e^t * cos(2t) ] from 0 to 𝜋/2 - ∫ from 0 to 𝜋/2 -1/4 * e^t * cos(2t) dt.

This integral can be computed similarly, and you will find that the result is a relatively simple expression.

Finally, divide this result by 𝜋/2 to find the average value of the function on the interval [0, 𝜋/2].