Final answer:
To solve the given problems, we need to find the definite integral of the given functions over their respective intervals. The first problem involves finding the definite integral of 3/(2x)^(2/3) from 1 to 8, which simplifies to 279/5. The second problem involves finding the definite integral of (x^7)/14 + 1/(10x^5) from 1 to 2, which simplifies to 30.1.
Step-by-step explanation:
To solve the problem 1. f(x) = 3/(2x)2/3 on [1,8], we can find the definite integral of f(x) over the interval [1,8].
- First, let's find the antiderivative of f(x). The antiderivative of xn is (1/(n+1))xn+1. Applying this rule, the antiderivative of 3/(2x)2/3 is (3/((2/3)+1))x^((2/3)+1), which simplifies to (3/(5/3))x^(5/3) or (9/5)x^(5/3).
- Next, we substitute the upper limit (8) and the lower limit (1) into the antiderivative expression and subtract the results. This gives us (9/5)(8^(5/3)) - (9/5)(1^(5/3)). Calculating further, we get (9/5)(32) - (9/5)(1), which simplifies to 288/5 - 9/5 = 279/5.
Therefore, the solution to the problem 1 is f(x) = 279/5.
For problem 2, f(x) = (x^7)/14 + 1/(10x^5) on [1,2]. Using the same approach, we find the antiderivative of f(x) to be (1/8)x^8 + (1/(-40))x^(-4), which simplifies to (1/8)x^8 - (1/40)x^(-4). Evaluating this antiderivative from 1 to 2, we get ((1/8)(2^8) - (1/40)(2^(-4))) - ((1/8)(1^8) - (1/40)(1^(-4))), which simplifies to (256/8 - 1/10) - (1/8 - 1/40) and further simplifies to 30.1.