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`