text: # Consider I= integrate from 3 to 9 of ((ln(9-x))/(ln(x-9)+ln(x-3))dx a)Show that integrate from a to b of f(x)dx=integrate from a to b of f(a+b-x)dx
  
  jeew-m | Certified Educator
  
  Let ;
  
  `intf(x)dx = g(x)`
  
  Then;
  
  int_a^b f(x)dx
  
  `= [g(x)]^b_a`
  
  `= g(b)-g(a)`
  
  `int_a^b f(x)dx = = g(b)-g(a) -----(1)`
  
  let;
  
  `t = a+b-x`
  
  `(dt)/dx = -1`
  
  ` -dt = dx`
  
  When x = a then t = b
  
  When x = b then t = a
  
  `int^b_a f(a+b-x)dx`
  
  `= int^a_b f(t)(-dt)`
  
  `= -int^a_b f(t)dt` This is same as `int_a^b f(x)dx` .
  
  `= -[g(t)]^a_b`
  
  `= -[g(a)-g(b)]`
  
  `= g(b)-g(a)`
  
  `int^b_a f(a+b-x)dx = g(b)-g(a) -----(2)`
  
  (1) = (2)
  
  Therefore;
  
  `int_a^b f(x)dx = int^b_a f(a+b-x)dx`
explanation: The document contains a math question and a detailed answer. The question is about proving a property of definite integrals, specifically showing that the integral of a function from a to b is equal to the integral of the function evaluated at (a+b-x) from a to b. The answer provides a step-by-step proof of this property.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Consider I= integrate from 3 to 9 of ((ln(9-x))/(ln(x-9)+ln(x-3))dx a)Show that integrate from a to b of f(x)dx=integrate from a to b of f(a+b-x)dx
  
  jeew-m | Certified Educator
  
  Let ;
  
  `intf(x)dx = g(x)`
  
  Then;
  
  int_a^b f(x)dx
  
  `= [g(x)]^b_a`
  
  `= g(b)-g(a)`
  
  `int_a^b f(x)dx = = g(b)-g(a) -----(1)`
  
  let;
  
  `t = a+b-x`
  
  `(dt)/dx = -1`
  
  ` -dt = dx`
  
  When x = a then t = b
  
  When x = b then t = a
  
  `int^b_a f(a+b-x)dx`
  
  `= int^a_b f(t)(-dt)`
  
  `= -int^a_b f(t)dt` This is same as `int_a^b f(x)dx` .
  
  `= -[g(t)]^a_b`
  
  `= -[g(a)-g(b)]`
  
  `= g(b)-g(a)`
  
  `int^b_a f(a+b-x)dx = g(b)-g(a) -----(2)`
  
  (1) = (2)
  
  Therefore;
  
  `int_a^b f(x)dx = int^b_a f(a+b-x)dx`
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and a detailed answer. The question is about proving a property of definite integrals, specifically showing that the integral of a function from a to b is equal to the integral of the function evaluated at (a+b-x) from a to b. The answer provides a step-by-step proof of this property.
  
  Label: question-and-answer