How do I solve this tricky calculus substitution question?

[chewy]

Homework Statement

if f is a continuous function, find the value of the integral

I = definte integral int [ f(x) / f(x) + f(a-x) ] dx from 0 to a. by making the substitution u = a - x and adding the resulting integral to I.

this is one of the last questions in the thomas international edition calculus on substitution, for the life of me i can't see how to do this, a pointer on where to attack this would be very helpful, as i can't let it go!

Homework Equations

The Attempt at a Solution

ive obviously used the substitution which was in the question, but it doesn't really help, as when i change the variable, the resulting integral is just as bad, and i don't know how adding it to the I helps, i substituted f(x) = x, into the integral and got the answer a/2, but i can't repeat it for any other random functions. I believe a/2 is the right answer but how i got it was not the right method. help

 

[Dick]

It's not that bad. It's just confusing. If you do the suggested substitution, your integrand is f(a-u)/(f(u)+f(a-u))*(-du). But now you should realize that instead of integrating x from 0 to a, you are integrating u from a to 0. So reverse the limits of integration introducing a - sign. This kills the minus sign on the du. Now do the harmless substitution u=x (it's just a dummy variable) and add it to your original integral. What's the integrand? What's your conclusion?

 

[Dick]

Oh, yeah, and welcome to the forums.

 

[chewy]

thanks very much for the reply, i followed your advice and the integrand i had was 1, which in turn leads to a when integrated, I am a little confused over the dummy variable, is u=x because the limits of the integrand are still [0-a] when reversed? also the answer in the book is a/2, does this mean i have to half the answer as i have added another integrand and if so how does this work as the new integrand is of a different value as the numerator is f(a-x) on the new one as opposed to f(x) on the original which i also verifed when i put f(x) = x^2 into the new integrand trying to understand how it actually works. or have i calculated wrong??

 

[chewy]

forget that last reply, i figured it out this morning when i woke with a fresh head. the graphs have the same area but the new one comes from the other side, when f(x) = 0, then f(a-x)= a, when f(x)= a the f(a-x) = 0. painfully obvious now.
this is website is a great idea, thanks again for your help

 

[Dick]

forget that last reply, i figured it out this morning when i woke with a fresh head. the graphs have the same area but the new one comes from the other side, when f(x) = 0, then f(a-x)= a, when f(x)= a the f(a-x) = 0. painfully obvious now.
this is website is a great idea, thanks again for your help

You've got it. You're welcome.