text: 1. Riemann Integration
  
  This is an integrable function. However, how is this integral solved? I can perfectly visualize this function. I don't see how the value of integral can be anything besides 0. But I cannot put in words or show work as to how the integral is solved. Please help
  
  2. I believe this what they call a improper integral, and because it is obviously undefined at 0, for something like this I believe you will have to do
  
  $\displaystyle \lim_{x\to0+}{\int_B}^1\frac{1}{x}$
  
  This should get you started
  
  3. What is B in the integral you entered? I don't understand that.
  
  4. Originally Posted by xyz
  
  This is an integrable function. However, how is this integral solved? I can perfectly visualize this function. I don't see how the value of integral can be anything besides 0. But I cannot put in words or show work as to how the integral is solved. Please help
  Note that over any subinterval $\displaystyle [a,b]\subset [0,1]$ that $\displaystyle \inf_{x\in[a,b]}f(x)=0$....so
  
  5. Is $\displaystyle f(x)=0$ everywhere with the exception of $\displaystyle x=\frac{1}{n}$ , $\displaystyle n \in \mathbb{N}$ so that you can devide the integration interval in subintervals of the type...
  
  $\displaystyle \frac{3}{4}<x\le 1$ , $\displaystyle \frac{5}{12} < x \le \frac{3}{4}$, $\displaystyle \dots$ , $\displaystyle \frac{n+\frac{3}{2}}{n+2}< x \le \frac{n+\frac{1}{2}}{n+1}$ , $\displaystyle \dots$
  
  ... so that in each interval is $\displaystyle f(x)=0$ with the only exception of $\displaystyle x=\frac{1}{n}$, i.e. in only one point. In each interval the function is Riemann integrable and is...
  
  $\displaystyle \int_{x_{n+1}}^{x_{n}} f(x)\cdot dx=0$ (1)
  
  From (1)...
  
  $\displaystyle \int_{0}^{1} f(x)\cdot dx = \sum_{n} \int_{x_{n+1}}^{x_{n}} f(x)\cdot dx=0$ (2)
  
  Kind regards
  
  $\displaystyle \chi$ $\displaystyle \sigma$
  
  6. Thanks a lot for all the help. I already got the idea from Drexel's hint. It was easy after that. But your solution is also very interesting!
explanation: The document contains a discussion about solving a specific integral, which is a math question. The initial post poses the question, and subsequent posts provide hints, clarifications, and a detailed solution. The final post acknowledges the help received and indicates that the problem has been solved.
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distilabel_metadata.content: 1. Riemann Integration
  
  This is an integrable function. However, how is this integral solved? I can perfectly visualize this function. I don't see how the value of integral can be anything besides 0. But I cannot put in words or show work as to how the integral is solved. Please help
  
  2. I believe this what they call a improper integral, and because it is obviously undefined at 0, for something like this I believe you will have to do
  
  $\displaystyle \lim_{x\to0+}{\int_B}^1\frac{1}{x}$
  
  This should get you started
  
  3. What is B in the integral you entered? I don't understand that.
  
  4. Originally Posted by xyz
  
  This is an integrable function. However, how is this integral solved? I can perfectly visualize this function. I don't see how the value of integral can be anything besides 0. But I cannot put in words or show work as to how the integral is solved. Please help
  Note that over any subinterval $\displaystyle [a,b]\subset [0,1]$ that $\displaystyle \inf_{x\in[a,b]}f(x)=0$....so
  
  5. Is $\displaystyle f(x)=0$ everywhere with the exception of $\displaystyle x=\frac{1}{n}$ , $\displaystyle n \in \mathbb{N}$ so that you can devide the integration interval in subintervals of the type...
  
  $\displaystyle \frac{3}{4}<x\le 1$ , $\displaystyle \frac{5}{12} < x \le \frac{3}{4}$, $\displaystyle \dots$ , $\displaystyle \frac{n+\frac{3}{2}}{n+2}< x \le \frac{n+\frac{1}{2}}{n+1}$ , $\displaystyle \dots$
  
  ... so that in each interval is $\displaystyle f(x)=0$ with the only exception of $\displaystyle x=\frac{1}{n}$, i.e. in only one point. In each interval the function is Riemann integrable and is...
  
  $\displaystyle \int_{x_{n+1}}^{x_{n}} f(x)\cdot dx=0$ (1)
  
  From (1)...
  
  $\displaystyle \int_{0}^{1} f(x)\cdot dx = \sum_{n} \int_{x_{n+1}}^{x_{n}} f(x)\cdot dx=0$ (2)
  
  Kind regards
  
  $\displaystyle \chi$ $\displaystyle \sigma$
  
  6. Thanks a lot for all the help. I already got the idea from Drexel's hint. It was easy after that. But your solution is also very interesting!
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distilabel_metadata.raw_output_q_a_detector_0: The document contains a discussion about solving a specific integral, which is a math question. The initial post poses the question, and subsequent posts provide hints, clarifications, and a detailed solution. The final post acknowledges the help received and indicates that the problem has been solved.
  
  Label: question-and-answer