Closed-form expression for an integral

Clash Royale CLAN TAG#URR8PPP

up vote
0
down vote

favorite

Let $a,b,c$ be positive real numbers. Let $x(t)$ denote the solution of the following differential equation
$$
dotx = a - bsin(x), quad x(0)inmathbbR.
$$

I'm interested in the computation of the following integral
$$
int_0^t cos(x(tau))e^ctau,mathrmdtau.
$$

In particular, is it possible to derive a closed-form expression of the above integral?

So far, I've tried a brute-force approach via Wolfram Alpha. I can find (a quite nasty) an explicit expression for $x(t)$, but, using this expression, the computation of the integral seems not feasible with this symbolic tool. So I'm wondering if there exists some elegant tricks, that can avoid the use of symbolic math softwares. Thanks for your help!

integration differential-equations analysis

share|cite|improve this question

edited Jul 18 at 20:47

asked Jul 18 at 18:17

Ludwig

737613

add a comment | 

up vote
0
down vote

favorite

Let $a,b,c$ be positive real numbers. Let $x(t)$ denote the solution of the following differential equation
$$
dotx = a - bsin(x), quad x(0)inmathbbR.
$$

I'm interested in the computation of the following integral
$$
int_0^t cos(x(tau))e^ctau,mathrmdtau.
$$

In particular, is it possible to derive a closed-form expression of the above integral?

So far, I've tried a brute-force approach via Wolfram Alpha. I can find (a quite nasty) an explicit expression for $x(t)$, but, using this expression, the computation of the integral seems not feasible with this symbolic tool. So I'm wondering if there exists some elegant tricks, that can avoid the use of symbolic math softwares. Thanks for your help!

integration differential-equations analysis

share|cite|improve this question

edited Jul 18 at 20:47

asked Jul 18 at 18:17

Ludwig

737613

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

Let $a,b,c$ be positive real numbers. Let $x(t)$ denote the solution of the following differential equation
$$
dotx = a - bsin(x), quad x(0)inmathbbR.
$$

I'm interested in the computation of the following integral
$$
int_0^t cos(x(tau))e^ctau,mathrmdtau.
$$

In particular, is it possible to derive a closed-form expression of the above integral?

So far, I've tried a brute-force approach via Wolfram Alpha. I can find (a quite nasty) an explicit expression for $x(t)$, but, using this expression, the computation of the integral seems not feasible with this symbolic tool. So I'm wondering if there exists some elegant tricks, that can avoid the use of symbolic math softwares. Thanks for your help!

integration differential-equations analysis

share|cite|improve this question

edited Jul 18 at 20:47

asked Jul 18 at 18:17

Ludwig

737613

Let $a,b,c$ be positive real numbers. Let $x(t)$ denote the solution of the following differential equation
$$
dotx = a - bsin(x), quad x(0)inmathbbR.
$$

I'm interested in the computation of the following integral
$$
int_0^t cos(x(tau))e^ctau,mathrmdtau.
$$

In particular, is it possible to derive a closed-form expression of the above integral?

So far, I've tried a brute-force approach via Wolfram Alpha. I can find (a quite nasty) an explicit expression for $x(t)$, but, using this expression, the computation of the integral seems not feasible with this symbolic tool. So I'm wondering if there exists some elegant tricks, that can avoid the use of symbolic math softwares. Thanks for your help!

integration differential-equations analysis

share|cite|improve this question

edited Jul 18 at 20:47

asked Jul 18 at 18:17

Ludwig

737613

share|cite|improve this question

share|cite|improve this question

edited Jul 18 at 20:47

edited Jul 18 at 20:47

edited Jul 18 at 20:47

asked Jul 18 at 18:17

Ludwig

737613

asked Jul 18 at 18:17

Ludwig

737613

asked Jul 18 at 18:17

Ludwig

737613

737613

add a comment | 

add a comment | 

1 Answer
1

active

oldest

votes

up vote
0
down vote

HINT, we have:

$$textyspace'left(tright)=texta-textbcdotsinleft(textyleft(tright)right)tag1$$

Divide both sides by the RHS:

$$fractextyspace'left(tright)texta-textbcdotsinleft(textyleft(tright)right)=1tag2$$

Integrate both sides with respect to $t$:

$$intfractextyspace'left(tright)texta-textbcdotsinleft(textyleft(tright)right)spacetextdt=int1spacetextdttag3$$

The RHS of equation $left(3right)$, equals:

$$int1spacetextdt=t+textC_1tag4$$

For the LHS of equation $left(3right)$, we substitute $textu:=textyleft(tright)$:

$$intfractextyspace'left(tright)texta-textbcdotsinleft(textyleft(tright)right)spacetextdt=intfrac1texta-textbcdotsinleft(texturight)spacetextdtextutag5$$

Now, substitute $texts:=tanleft(fractextu2right)$.

---

Then you end up with:

$$intfrac1texta-textbcdotsinleft(texturight)spacetextdtextu=2cdotintfrac1left(textscdotsqrttexta-fractextbsqrttextaright)^2+fractexta^2-textb^2textaspacetextdtextstag6$$

share|cite|improve this answer

answered Jul 18 at 18:32

Jan

21.6k31239

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  The question is about the integral involving $x(t)$, not necessarily the closed form of $x(t)$ itself
  – Dylan
  Jul 18 at 20:02
  

  
  
  
  

add a comment | 

Your Answer

StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: false,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);

 

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2855857%2fclosed-form-expression-for-an-integral%23new-answer', 'question_page');

);

Post as a guest

Name Email

1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
0
down vote

HINT, we have:

$$textyspace'left(tright)=texta-textbcdotsinleft(textyleft(tright)right)tag1$$

Divide both sides by the RHS:

$$fractextyspace'left(tright)texta-textbcdotsinleft(textyleft(tright)right)=1tag2$$

Integrate both sides with respect to $t$:

$$intfractextyspace'left(tright)texta-textbcdotsinleft(textyleft(tright)right)spacetextdt=int1spacetextdttag3$$

The RHS of equation $left(3right)$, equals:

$$int1spacetextdt=t+textC_1tag4$$

For the LHS of equation $left(3right)$, we substitute $textu:=textyleft(tright)$:

$$intfractextyspace'left(tright)texta-textbcdotsinleft(textyleft(tright)right)spacetextdt=intfrac1texta-textbcdotsinleft(texturight)spacetextdtextutag5$$

Now, substitute $texts:=tanleft(fractextu2right)$.

---

Then you end up with:

$$intfrac1texta-textbcdotsinleft(texturight)spacetextdtextu=2cdotintfrac1left(textscdotsqrttexta-fractextbsqrttextaright)^2+fractexta^2-textb^2textaspacetextdtextstag6$$

share|cite|improve this answer

answered Jul 18 at 18:32

Jan

21.6k31239

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  The question is about the integral involving $x(t)$, not necessarily the closed form of $x(t)$ itself
  – Dylan
  Jul 18 at 20:02
  

  
  
  
  

add a comment | 

up vote
0
down vote

HINT, we have:

$$textyspace'left(tright)=texta-textbcdotsinleft(textyleft(tright)right)tag1$$

Divide both sides by the RHS:

$$fractextyspace'left(tright)texta-textbcdotsinleft(textyleft(tright)right)=1tag2$$

Integrate both sides with respect to $t$:

$$intfractextyspace'left(tright)texta-textbcdotsinleft(textyleft(tright)right)spacetextdt=int1spacetextdttag3$$

The RHS of equation $left(3right)$, equals:

$$int1spacetextdt=t+textC_1tag4$$

For the LHS of equation $left(3right)$, we substitute $textu:=textyleft(tright)$:

$$intfractextyspace'left(tright)texta-textbcdotsinleft(textyleft(tright)right)spacetextdt=intfrac1texta-textbcdotsinleft(texturight)spacetextdtextutag5$$

Now, substitute $texts:=tanleft(fractextu2right)$.

---

Then you end up with:

$$intfrac1texta-textbcdotsinleft(texturight)spacetextdtextu=2cdotintfrac1left(textscdotsqrttexta-fractextbsqrttextaright)^2+fractexta^2-textb^2textaspacetextdtextstag6$$

share|cite|improve this answer

answered Jul 18 at 18:32

Jan

21.6k31239

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  The question is about the integral involving $x(t)$, not necessarily the closed form of $x(t)$ itself
  – Dylan
  Jul 18 at 20:02
  

  
  
  
  

add a comment | 

up vote
0
down vote

up vote
0
down vote

HINT, we have:

$$textyspace'left(tright)=texta-textbcdotsinleft(textyleft(tright)right)tag1$$

Divide both sides by the RHS:

$$fractextyspace'left(tright)texta-textbcdotsinleft(textyleft(tright)right)=1tag2$$

Integrate both sides with respect to $t$:

$$intfractextyspace'left(tright)texta-textbcdotsinleft(textyleft(tright)right)spacetextdt=int1spacetextdttag3$$

The RHS of equation $left(3right)$, equals:

$$int1spacetextdt=t+textC_1tag4$$

For the LHS of equation $left(3right)$, we substitute $textu:=textyleft(tright)$:

$$intfractextyspace'left(tright)texta-textbcdotsinleft(textyleft(tright)right)spacetextdt=intfrac1texta-textbcdotsinleft(texturight)spacetextdtextutag5$$

Now, substitute $texts:=tanleft(fractextu2right)$.

---

Then you end up with:

$$intfrac1texta-textbcdotsinleft(texturight)spacetextdtextu=2cdotintfrac1left(textscdotsqrttexta-fractextbsqrttextaright)^2+fractexta^2-textb^2textaspacetextdtextstag6$$

share|cite|improve this answer

answered Jul 18 at 18:32

Jan

21.6k31239

HINT, we have:

$$textyspace'left(tright)=texta-textbcdotsinleft(textyleft(tright)right)tag1$$

Divide both sides by the RHS:

$$fractextyspace'left(tright)texta-textbcdotsinleft(textyleft(tright)right)=1tag2$$

Integrate both sides with respect to $t$:

$$intfractextyspace'left(tright)texta-textbcdotsinleft(textyleft(tright)right)spacetextdt=int1spacetextdttag3$$

The RHS of equation $left(3right)$, equals:

$$int1spacetextdt=t+textC_1tag4$$

For the LHS of equation $left(3right)$, we substitute $textu:=textyleft(tright)$:

$$intfractextyspace'left(tright)texta-textbcdotsinleft(textyleft(tright)right)spacetextdt=intfrac1texta-textbcdotsinleft(texturight)spacetextdtextutag5$$

Now, substitute $texts:=tanleft(fractextu2right)$.

---

Then you end up with:

$$intfrac1texta-textbcdotsinleft(texturight)spacetextdtextu=2cdotintfrac1left(textscdotsqrttexta-fractextbsqrttextaright)^2+fractexta^2-textb^2textaspacetextdtextstag6$$

share|cite|improve this answer

answered Jul 18 at 18:32

Jan

21.6k31239

share|cite|improve this answer

share|cite|improve this answer

answered Jul 18 at 18:32

Jan

21.6k31239

answered Jul 18 at 18:32

Jan

21.6k31239

answered Jul 18 at 18:32

Jan

21.6k31239

21.6k31239

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  The question is about the integral involving $x(t)$, not necessarily the closed form of $x(t)$ itself
  – Dylan
  Jul 18 at 20:02
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  The question is about the integral involving $x(t)$, not necessarily the closed form of $x(t)$ itself
  – Dylan
  Jul 18 at 20:02
  

  
  
  
  

1

1

The question is about the integral involving $x(t)$, not necessarily the closed form of $x(t)$ itself
– Dylan
Jul 18 at 20:02

The question is about the integral involving $x(t)$, not necessarily the closed form of $x(t)$ itself
– Dylan
Jul 18 at 20:02

add a comment | 

 

draft saved

draft discarded

 

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2855857%2fclosed-form-expression-for-an-integral%23new-answer', 'question_page');

);

Post as a guest

Name Email

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

Name Email

Name

Email

Name Email

Name

Email