Define:
Partition $P = \{x_0,\dots,x_n\}$
$L(P) = \sum_{i=1}^n (\alpha(x_i)-\alpha(x_{i-1}) \inf (f(x): x \in [x_{i-1}, x_i])$
$U(P) = \sum_{i=1}^n (\alpha(x_i)-\alpha(x_{i-1}) \sup (f(x): x \in [x_{i-1}, x_i])$
$L(P) \le I \le U(P)$. I is the Riemann-Stieltjes Integral.
Using this definition, and assume f(x) is a continues function then: 1) $\int_0^1 f(x)2^{-[x]} d(2^{[x]}) = 0.5f(1)$ 2) $\int_1^2 f(x) d(x[x+3]) = 4 \int_1^2 f(x) dx + 2f(2)$
My attempt to 1)
$\sum_{i=1}^n 2^{[x_i]} – 2^{[x_{i-1}]} = 2^1 – 2^0 = 1$
$L(P) = 1 \cdot \inf (f(x)2^{[-x]}) = 1 \cdot 2^0f(0) = f(0)$
$U(P) = 1 \cdot \sup (f(x)2^{[-x]}) = 1 * 2^{-1}f(1) = 0.5f(1)$
Clearly my L(P) is wrong but I’m not sure how/why? And I’m completely stuck on the second question. Can someone please help me? I may be using the definition incorrect. NB: I know nothing about the extreme value theorem etc. Basically no other knowledge apart from this definition! Lol
Question 2: If $f’(x)$ is continuous show $2 \int_a^b f(x)d(f(x)[x]) = [f^2(x)[x]] (a,b) + \int_a^b f^2(x)d[x]$
My attempted solution: Using integration by parts: 2(fx)f(x)[x] between (a,b) – int (a,b) f(x)[x]df(x) = 2(fx)f(x)[x] between (a,b) – int (a,b) f(x)[x]f’(x)dx Not sure how to get that second term!
Question 3: How do you start to calculate these: $\int_1^n x^3 d([x]/x^2)$
$\int_1^\infty d([\log x]/x^2)$
$\int_1^n x(x-1) d(1/[x])$