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## #1 2009-09-14 09:58:19

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,123

### Integral and inequality

Hi;

Evaluate this integral:

Prove the inequality:

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #2 2009-09-14 10:12:44

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

### Re: Integral and inequality

Are we allowed to express the LHS of the inequality as a Taylor series? If so, the problem becomes trivial.

Why did the vector cross the road?
It wanted to be normal.

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## #3 2009-09-14 10:28:28

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,123

### Re: Integral and inequality

Hi mathsyperson;

Thats how I would do it. But in this Putnam problem book he doesn't use that method. Why, I can't say.

Last edited by bobbym (2009-09-14 11:42:02)

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

## #4 2009-09-14 12:43:51

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

### Re: Integral and inequality

Ever seen this theorem?

Let
be convex and differentiable. Then

[align=center]

[/align]

So you just put

and
.

(What Im saying is that no point on a convex and differentiable curve lies below the corresponding point on a straight line that is tangent to any part of the curve.)

PS: I just looked up Wikipedia. This theorem is stated there without proof.

Last edited by JaneFairfax (2009-09-14 12:56:36)

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## #5 2009-09-14 13:07:40

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

### Re: Integral and inequality

mathsyperson wrote:

Are we allowed to express the LHS of the inequality as a Taylor series? If so, the problem becomes trivial.

bobbym wrote:

Thats how I would do it.

The inequality holds for

as well, in which case the method will not be trivial for
.

And note that there is an apostrophe in Thats.

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## #6 2009-09-14 16:21:12

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,123

### Re: Integral and inequality

Hello Jane;

Jane wrote:

The inequality holds for

as well, in which case the method will not be trivial for
.

But the problem states x > 0 and mathsy is right using the expansion of e^x makes it trivial.

Jane wrote:

And note that there is an apostrophe in Thats.

That is true Jane, poor typing and high speeds, a lethal combination. Thanks for the correction.

Jane wrote:

Ever seen this theorem?

Yes Jane, I have heard of that theorem but I forgot it.

Last edited by bobbym (2009-09-14 18:23:55)

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #7 2009-09-15 07:36:20

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,123

### Re: Integral and inequality

Hi;

The inequality was the easy one. Now how about the integral it is even easier. If you start substituting you will go mad.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #8 2009-09-25 00:11:45

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,123

### Re: Integral and inequality

Hi;

Another integral this one is also easy:

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #9 2009-09-25 01:06:15

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

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## #10 2009-09-25 11:13:48

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,123

### Re: Integral and inequality

Hi Jane;

Last edited by bobbym (2009-09-25 11:14:14)

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

## #11 2010-09-28 07:36:42

123ronnie321
Member
Registered: 2010-09-28
Posts: 128

### Re: Integral and inequality

bobbym wrote:

Hi;

Another integral this one is also easy:

(e^2x)/2 - (e^-2x)/2

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## #12 2010-09-28 07:41:17

123ronnie321
Member
Registered: 2010-09-28
Posts: 128

### Re: Integral and inequality

bobbym wrote:

Hi;

Evaluate this integral:

Prove the inequality:

1/2[x - log(sinx + cosx)]

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## #13 2010-09-28 08:24:31

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,123

### Re: Integral and inequality

Hi 123ronnie321;

Yes that is almost correct!

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #14 2011-03-29 23:05:19

gAr
Member
Registered: 2011-01-09
Posts: 3,479

### Re: Integral and inequality

Hi bobbym,

Evaluate this integral:

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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## #15 2011-03-29 23:49:54

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,123

### Re: Integral and inequality

Hi gAr;

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #16 2011-03-30 00:07:52

gAr
Member
Registered: 2011-01-09
Posts: 3,479

### Re: Integral and inequality

Thank you!

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Offline

## #17 2011-03-30 00:13:01

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,123

### Re: Integral and inequality

HI gAr;

You are welcome.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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