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

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jeew-m | College Teacher | (Level 1) Educator Emeritus

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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`

 

Sources:

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