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.
Expert Answers
jeew-m
| Certified Educator
| Certified Educator
calendarEducator since 2012
write1,657 answers
starTop subjects are Math, Science, and Social Sciences
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`
Related Questions
- Evaluate the integral [ln(x)/(x) dx]
- 1 Educator Answer
- Consider I= integrate from 3 to 9 of ((ln(9-x))/(ln(x-9)+ln(x-3))dx a)Show that integrate from a...
- 1 Educator Answer
- Consider I= integrate of ((ln(9-x))/(ln(x-9)+ln(x-3))dx a)Show that integrate from a to b of...
- 1 Educator Answer
- Evaluate the indefinite integral integrate of (x^3(ln(x))dx)
- 2 Educator Answers
- Consider I= integrate from 3 to 9 of ((ln(9-x))/(ln(x-9)+ln(x-3))dx a)Show that integrate...
- 1 Educator Answer