Consider I= integrate from 3 to 9 of ((ln(9-x))/(ln(x-9)+ln(x-3))dx
Answer part a
a)Show that integrate from a to b of f(x)dx=integrate from a to b of f(a+b-x)dx
b) Use the above property to write I with the same limits of integration but a different integrand.
c) Add both expressions for I that you now have and use the result to evaluate it.
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`intf(x)dx = g(x)`
`int_a^b f(x)dx `
`int_a^b f(x)dx = = g(b)-g(a) -----(1)`
`t = a+b-x`
`(dt)/dx = -1`
` -dt = dx`
When x = a then t = b
When x = b then t = a
`= int^a_b f(t)(-dt)`
`= -int^a_b f(t)dt` This is same as `int_a^b f(x)dx` .
`int^b_a f(a+b-x)dx = g(b)-g(a) -----(2)`
(1) = (2)
`int_a^b f(x)dx = int^b_a f(a+b-x)dx`
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