# 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`
The text passage contains a mathematical question about proving an integral property and provides a detailed answer to that question.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 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`
user
The text passage contains a mathematical question about proving an integral property and provides a detailed answer to that question.

TYPE: SINGLE_QUESTION_WITH_ANSWER