Integration by parts - In need of desperate help!!!

[bubbles930]

Let F be a non-decreasing function that is smooth everywhere on its domain [0, ?] ⊆ ℝ+.
Let f denote the derivative of F (that is, ? = ?′). Since F is non-decreasing, ?(?) ≥ 0 at every x.
Suppose that ?(0) = 0 and ?(?) = 1.

Use integration by parts to show that:



Can someone PLEASE help me work this out?

I have found this formula to help if you need.

 

[Steven G]

1st of all if F(x)=1 then how can F(0)=1

More importantly what have you tried, where are you stuck? I assume that you tried to solve one or both of the integrals by parts. Can we see your work?

You do realize that this is a math help forum where we expect students to solve their own problems with the assistance of forum helpers as opposed to just doing the problem for the student. After all, what kind of help would that be?

Please post back with your work so we know how to help you.

 

[bubbles930]

1st of all if F(x)=1 then how can F(0)=1

More importantly what have you tried, where are you stuck? I assume that you tried to solve one or both of the integrals by parts. Can we see your work?

You do realize that this is a math help forum where we expect students to solve their own problems with the assistance of forum helpers as opposed to just doing the problem for the student. After all, what kind of help would that be?

Please post back with your work so we know how to help you.

Hi Jomo,

Thanks for your reply. I realise that providing no work to show on this thread is frustrating. The fact is, I am really struggling with this problem and do not know where to start. Do you think you could get me a tip on where/how to tackle this problem? I'd appreciate it hugely. Cheers.

 

[Deleted member 4993]

Let F be a non-decreasing function that is smooth everywhere on its domain [0, ?] ⊆ ℝ+.
Let f denote the derivative of F (that is, ? = ?′). Since F is non-decreasing, ?(?) ≥ 0 at every x.
Suppose that ?(0) = 0 and ?(?) = 1.

Use integration by parts to show that:

View attachment 18773

Can someone PLEASE help me work this out?

I have found this formula to help if you need.

View attachment 18774

View attachment 18774

start by making table of all the parts in proposition 8.1 and their equivalent in your problem. e.g.

x in proposition 8.1 is 0 in the given problem

g(x) in proposition 8.1 is x in the given problem ........ continue

This will give you better grasp of the process you need to follow.

 

[HallsofIvy]

Jomo, the limits of integration are 0 and \(\displaystyle \overline{x}\). I believe the OP meant \(\displaystyle F(\overline{x})= 1\), not F(x)= 1.

Bubbles930, using "integration by parts", \(\displaystyle \int_0^{\overline{x}} udv= \left[uv\right]_0^{\overline{x}}- \int_0^{\overline{x}} v du\), let u= x and \(\displaystyle dv= f(x)dx\). Then \(\displaystyle du= dx\) and \(\displaystyle v= F(x)\). So we have \(\displaystyle \left[xF(x)\right]_0^{\overline{x}}- \int_0^{\overline{x}} F(x)dx\).

Since \(\displaystyle F(\overline{x})= 1\) that is \(\displaystyle \overline{x}- \int_0^{\overline{x}} F(x)dx\).

Further, since \(\displaystyle \int_0^{\overline{x}} 1 dx= \overline{x}\) that is​

\(\displaystyle \int_0^{\overline{x}} 1- F(x) dx\).​

 

[bubbles930]

Jomo, the limits of integration are 0 and \(\displaystyle \overline{x}\). I believe the OP meant \(\displaystyle F(\overline{x})= 1\), not F(x)= 1.

Bubbles930, using "integration by parts", \(\displaystyle \int_0^{\overline{x}} udv= \left[uv\right]_0^{\overline{x}}- \int_0^{\overline{x}} v du\), let u= x and \(\displaystyle dv= f(x)dx\). Then \(\displaystyle du= dx\) and \(\displaystyle v= F(x)\). So we have \(\displaystyle \left[xF(x)\right]_0^{\overline{x}}- \int_0^{\overline{x}} F(x)dx\).

Since \(\displaystyle F(\overline{x})= 1\) that is \(\displaystyle \overline{x}- \int_0^{\overline{x}} F(x)dx\).

Further, since \(\displaystyle \int_0^{\overline{x}} 1 dx= \overline{x}\) that is​

\(\displaystyle \int_0^{\overline{x}} 1- F(x) dx\).​

Thank you so much, this is great help!