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#1 2006-08-27 14:00:29

Zhylliolom
Real Member
Registered: 2005-09-05
Posts: 412

Integration by Substitution

Evaluate the following integrals by the method of integration by substitution. Starred exercises may take more problem solving and manipulation than the others. Double starred problems should only be attempted by those who are quite experienced with the Calculus, and may cause anger and frustration. Triple starred problems are reserved for the truly insanely skilled. The beauty of the solution of triple starred problems combined with the sense of accomplishment is a true reward for the hard work put into the problem.



Last edited by Zhylliolom (2006-08-28 10:32:46)

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#2 2006-08-27 14:50:07

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Integration by Substitution


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#3 2006-08-27 15:27:02

Zhylliolom
Real Member
Registered: 2005-09-05
Posts: 412

Re: Integration by Substitution

Good work Ricky, but as you must know, no credit is given for stating only the answer and not the solution method.

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#4 2006-08-27 15:45:29

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: Integration by Substitution

Ugh, you're going to make me type all that up?  Fine...

Since 3^{-4x^2} is symmetric across the y-axis.

Let:

Then:

Changing to polar coordinates:

Standard integration follows, we get:

Since 1/81 < 1, 1/81^a^2 approaches 0 as a approaches infinity.

So:

So the answer is:


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#5 2006-08-27 16:02:39

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Integration by Substitution

Wait a minute, I never had to use integration by parts...


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#6 2006-08-27 16:32:21

Zhylliolom
Real Member
Registered: 2005-09-05
Posts: 412

Re: Integration by Substitution

substitution*

Monumental effort, Ricky, and your solution is indeed correct. However, you are right in noticing that you didn't ever use any substitution. The solution takes much less work with the proper cleverness of substitution and manipulation and the right knowledge.

Last edited by Zhylliolom (2006-08-29 09:15:45)

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#7 2009-07-03 16:42:13

glenn101
Member
Registered: 2008-04-02
Posts: 108

Re: Integration by Substitution

1. Let u = 1+x^2
         du= 2x dx
= ∫√u du
= u^3/2
    ------
       2
=   2(1+x^2)^3/2
    ----------------    + c
          3

9. = ∫sinx/cosx dx
  let u = cosx
      du = -sinx dx
      =-∫1/u du
=     -loge(|cosx|) + c

Last edited by glenn101 (2009-07-03 16:44:00)


"If your going through hell, keep going."

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#8 2009-07-23 23:34:26

juriguen
Member
Registered: 2009-07-05
Posts: 59

Re: Integration by Substitution

Hi!

Lets try 17.

where the substitutions used have been z = - ln(x) first, and t^2 = z next.

Jose

Last edited by juriguen (2009-07-23 23:35:08)


“Make everything as simple as possible, but not simpler.” -- Albert Einstein

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#9 2009-07-23 23:50:05

juriguen
Member
Registered: 2009-07-05
Posts: 59

Re: Integration by Substitution

I found 16 is indeed much easier this way:

where for the first step I have used x = exp(ln(x)), and the substitution is then 4x^2 ln(3) = t^2. Finally, the last step is done evaluating the erf function.

Jose


“Make everything as simple as possible, but not simpler.” -- Albert Einstein

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#10 2009-07-24 01:44:07

juriguen
Member
Registered: 2009-07-05
Posts: 59

Re: Integration by Substitution

Now 18:

using 2 - x = z^2 !


“Make everything as simple as possible, but not simpler.” -- Albert Einstein

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#11 2009-07-24 03:45:00

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,998

Re: Integration by Substitution

For #15

Start with the first one and say:





Now for the second integral:

Say:



So

Last edited by bobbym (2009-07-24 04:06:21)


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#12 2009-07-24 04:01:11

juriguen
Member
Registered: 2009-07-05
Posts: 59

Re: Integration by Substitution

I would like to propose another integral, which took me really long to solve! (I would grade it at least with **)

If anyone feels this should be another post, since both integration by parts and substitution need to be used, just let me know smile

Otherwise, have fun with the problem. Here we go:

where

Enjoy!


“Make everything as simple as possible, but not simpler.” -- Albert Einstein

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#13 2009-07-24 05:51:57

juriguen
Member
Registered: 2009-07-05
Posts: 59

Re: Integration by Substitution

Zhylliolom


I am trying to figure out the last integral, but still struggling to understand a few things:

When you define J(x) you use p, which is any prime? and 1/n... But with the hint the summation changes and has ln(p)...

Also, in the integral to solve, I see dJ(x)... Does this mean that first we should find the differential of J(x)?


Thanks!


“Make everything as simple as possible, but not simpler.” -- Albert Einstein

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#14 2009-07-25 04:33:33

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,998

Re: Integration by Substitution

Three easy ones;

Last edited by bobbym (2009-07-26 03:38:25)


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#15 2009-07-25 19:46:35

juriguen
Member
Registered: 2009-07-05
Posts: 59

Re: Integration by Substitution

This is the only way I see for 19, but it is a little weird!


First consider:

where the substitution used has been alpha x^2 = u^2


Then,


Jose

Last edited by juriguen (2009-07-25 23:08:29)


“Make everything as simple as possible, but not simpler.” -- Albert Einstein

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#16 2009-07-29 12:14:36

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,998

Re: Integration by Substitution

Hi;

Here is an interesting integral done by substitution.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#17 2009-08-12 23:09:02

Identity
Member
Registered: 2007-04-18
Posts: 934

Re: Integration by Substitution

And another one:

Last edited by Identity (2009-08-12 23:11:56)

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#18 2009-08-13 23:08:34

rzaidan
Member
Registered: 2009-08-13
Posts: 59

Re: Integration by Substitution

Hi Identity
                                                              sec x + tan x
∫ (1/cos x) dx=∫ sec x dx=∫ sec x ______________      dx
                                                              sec x + tan x

                                       (sec x)^2 + sec x tan x
                               ∫      ___________________  dx
                                          sec x + tan x
   = ln(sec x + tan x) + C since the numerator is the derivative of the denominator.
Best Regards
Riad Zaidan

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#19 2009-08-14 02:57:44

Identity
Member
Registered: 2007-04-18
Posts: 934

Re: Integration by Substitution

Nice solution rzaidan, but I think the step of multiplying by (secx + tanx) requires a big leap of faith whereas u-substitution does not.

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#20 2009-08-15 02:12:30

rzaidan
Member
Registered: 2009-08-13
Posts: 59

Re: Integration by Substitution

Hi Identity
  This can be noticed from the derivative of ln(sec x + tan x) + C which is
                                              sec x tan x + sec^2 x      sec x ( tan x + sec x )
(d/dx) (ln(sec x + tan x) + C)=__________________  =  ___________________  =  sec x
                                                  secx  + tan x                    sec x + tan x

Best Regards
Riad Zaidan

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#21 2009-11-23 19:03:37

Denominator
Member
Registered: 2009-11-23
Posts: 155

Re: Integration by Substitution

Hi. This is my first time posting here.
I'm a year 7 and i think i know the answer to question number 1 and 8!

8 is 1/5 (e^5x)

1 is 2/3 (1+x^2)^(3/2)

May I ask how did you actually get the real forat of the equations? Did you copy from Microsoft Word Equations?

And I also figured out number 6!!!
Is it (1/3)sqrt(2x+1) ?

Please tell me if I'm right

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#22 2009-11-23 19:10:55

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 88,998

Re: Integration by Substitution

Hi Denominator;

Welcome to the forum!

I am sorry but that is not correct for number 6:

See my post # 14 above and click #6 for my answer.

Last edited by bobbym (2009-11-23 19:12:33)


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#23 2009-11-24 05:53:34

Denominator
Member
Registered: 2009-11-23
Posts: 155

Re: Integration by Substitution

Okay my bad.
How do you make the actual equation insetad of just typing it up in text?

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#24 2009-11-24 10:50:15

Identity
Member
Registered: 2007-04-18
Posts: 934

Re: Integration by Substitution

Ooops tongue sorry rzaidan I was going to post my solution but I forgot

Last edited by Identity (2009-11-24 10:50:48)

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#25 2010-09-28 07:13:18

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

Re: Integration by Substitution

bobbym wrote:

Hi;

Here is an interesting integral done by substitution.

Ans - 0

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