I'm having trouble solving this problem: definite integral (2-t)(sqrt(t)), 0..2

To solve the definite integral ∫(2-t)√(t) dt from 0 to 2, we can break it down into two separate integrals and then evaluate them individually.

First, let's separate the integral into two parts using the distributive property:

∫(2-t)√(t) dt = ∫2√(t) dt - ∫t√(t) dt

Now, let's solve each of these integrals separately.

1. ∫2√(t) dt:
To integrate this, we can use the power rule for integration, which is ∫x^n dx = (x^(n+1))/(n+1).

Applying this rule, we have:
∫2√(t) dt = 2 * ∫t^(1/2) dt = 2 * (t^(1/2 + 1))/(1/2 + 1) = 2 * (t^(3/2))/(3/2 + 1) = 4/3 * t^(3/2)

2. ∫t√(t) dt:
To integrate this, we can use the power rule again:
∫t√(t) dt = ∫t^(3/2) dt = (t^(3/2 + 1))/(3/2 + 1) = (t^(5/2))/(5/2)

Now, let's evaluate the two integrals from 0 to 2:

Using the upper limit of integration (2):
∫(2-t)√(t) dt = [4/3 * t^(3/2)] from 0 to 2 - [(t^(5/2))/(5/2)] from 0 to 2

Substituting the upper limit:
= [4/3 * (2)^(3/2)] - [(2^(5/2))/(5/2)]

Next, substituting the lower limit (0), we get:
= [4/3 * (2)^(3/2)] - [(2^(5/2))/(5/2)] - [4/3 * (0)^(3/2)] + [(0)^(5/2))/(5/2)]

Since any number raised to the power of 0 is 1, the last two terms become 0. Hence, we have:
= [4/3 * (2)^(3/2)] - [(2^(5/2))/(5/2)]

Now, let's simplify and calculate the numerical value to get the final result.