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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: $\int_1^n x^3 d([x]/x^2)$

$\int_1^\infty d([\log x]/x^2)$

$\int_1^n x(x-1) d(1/[x])$

  • 3
    @user10866: Please present your question in a more readable form. If you know LaTeX then please edit your question.2011-05-13
  • 0
    Do some people really use the notation $d(2^{[x]})$? The OP could take care to explain its meaning.2011-05-13
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    @user10866: Have a look at my edits - you will see how to write integral and sum. You can make your post readable.2011-05-13
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    @Didier: The original post contained things like "Integral (0,1) f(x)2^-[x] d(2^[x]) = 0.5f(1)". I was not sure, what [x] is supposed to mean, so I've kept it. I hope the OP will now be able to finish editing himself.2011-05-13
  • 0
    $P$ is a partition of interval $[a,b]$. Notation, $I = \int_a^b f(x)\,d\alpha(x)$. Also, some quantifiers and such "trivialities" are missing. I assume $[x]$ is the floor (or greatest integer) function.2011-05-13
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    @Martin: Yes. $ $2011-05-14

1 Answers 1