An integral by parts problem - please advise

[insane0hflex]

Homework Statement

The function is increasing and has a inverse f^-1
Also assume f′is continuous and f'(x) > 0 over the state interval of integration [a,b]

PLEASE NOTE! a is lower limit, b is upper limit (same for alpha and beta symbol later on)

Used integration by parts to show that:
[itex]\int f(x)dx=bf(b)−af(a)−∫ba xf′(x)dx[/itex]

Then if y = f(x), then the following is true (used the substitution rule)

[itex] \int _a ^b f(x) \: dx = bf(b) - af(a) - \int _{f(a)} ^{f(b)} f ^{-1} (y) \: dy [/itex]

Homework Equations

The question: Now, I need to show that if α = f(a), and β = f(b), then

[itex]∫βαf^-1(x)dx=β f^-1(β)−α f^-1(α)−∫f^-1(β)f^-1(α)f(x)dx[/itex]

The Attempt at a Solution

trying to fix the problem's display. B should be the upper limit, a should be lower limit when next to an integral sign

Im lost. I'm sure its something relatively easy to do, maybe another substitution or relevant manipulation?

Help appreciated!

 

[LCKurtz]

Homework Statement

The function is increasing and has a inverse f^-1
Also assume f′is continuous and f'(x) > 0 over the state interval of integration [a,b]

PLEASE NOTE! a is lower limit, b is upper limit (same for alpha and beta symbol later on)

Used integration by parts to show that:
[itex]\int f(x)dx=bf(b)−af(a)−∫ba xf′(x)dx[/itex]

It isn't true. Try f(x) = x, a = 1, b = 2.

 

[SammyS]

Homework Statement

The function is increasing and has a inverse f^-1
Also assume f′is continuous and f'(x) > 0 over the state interval of integration [a,b]

PLEASE NOTE! a is lower limit, b is upper limit (same for alpha and beta symbol later on)

Used integration by parts to show that:
[itex]\int f(x)dx=bf(b)−af(a)−\int_{\,b}^{\,a} xf'(x)dx[/itex]

Then if y = f(x), then the following is true (used the substitution rule)

[itex] \int _a ^b f(x) \: dx = bf(b) - af(a) - \int _{f(a)} ^{f(b)} f ^{-1} (y) \: dy [/itex]

Homework Equations

The question: Now, I need to show that if α = f(a), and β = f(b), then

[itex]\displaystyle \int_β^αf^{-1}(x)dx=β f^{-1}(β)−α f^-1(α)−\int_{f^{-1}(β)}^{f^{-1}(α)}f(x)dx[/itex]

The Attempt at a Solution

trying to fix the problem's display. B should be the upper limit, a should be lower limit when next to an integral sign

I'm lost. I'm sure its something relatively easy to do, maybe another substitution or relevant manipulation?

Help appreciated!

You have a few (apparent) typos, which I attempted to fix.

You have worked out most of the result.

(To view the LaTeX code, right click the desired expression and choose "Show Math As: TEX Commands".)