- #1
s3a
- 818
- 8
Homework Statement
Problem:
If f'(x) = 10t / ∛(t – 2) and f(8) = –20, calculate f(x).
Solution:
Let u = t – 2 ⇒ dx = du. Then f(x) = –20 + ∫_8^x [10t / ∛(t – 2)] dt = –20 + ∫_6^(x – 2) [10(u + 2) / ∛(u)] du = –20 + 10 ∫_6^(x – 2) [u^(2/3) + 2u^(–1/3)] du = 30 ∛[(x – 2)^2] + 6(x – 2)^(5/3) – 66 ∛(3) ∛(12) – 20
Additionally, the problem is attached as TheProblem.png, and the solution is attached as TheSolution.png.
Homework Equations
I'm not sure, but I think this has to do with the Fundamental Theorem of Calculus.
The Attempt at a Solution
I understand all the algebraic manipulations done; I'm just confused as to how the author went from the problem to the expression f(x) = –20 + ∫_8^x [10t / ∛(t – 2)] dt. Also, is it okay/valid that f'(x) (which is a function of x) = 10t / ∛(t – 2) (which is a function of t)?
Any help in clearing my confusions would be greatly appreciated!