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Evaluate $int frac12+sin x+cos xdx.$ [duplicate]

1

$begingroup$

This question already has an answer here:

• 
  What is the right first step to take when integrating $1/(2+ sin x + cos x)$? [duplicate]
  
  2 answers
  
  


• 
  Evaluating $int P(sin x, cos x) textdx$
  
  3 answers
  
  

$$int frac12+sin x+cos xdx.$$

My attempts:

1. Let $y = sin x+cos x.implies fracdydx=cos x-sin x=y'.$
   

$$int frac12+sin x+cos xdx=intfrac12+yfracdyy'.$$

And I tried to use the fact $(ln x)'=1/x,$ but $(ln(2+y))'=fracy'2+y$ : the form doesn't match. So I think I've failed at this moment.

2. Let $u=sin x.implies du=cos x dx=sqrt1-u^2dx.$

$$int frac12+sin x+cos xdx=intfrac12+u+sqrt1-u^2fracdusqrt1-u^2.$$

And it looks more uncomputable.

Both of my attempts are at a dead end. How to evaluate this integral? Any help would be appreciated.

calculus integration indefinite-integrals

share|cite|improve this question

asked Mar 25 at 7:51

user642721user642721

735

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marked as duplicate by Martin R, Robert Z calculus
Users with the  calculus badge can single-handedly close calculus questions as duplicates and reopen them as needed.

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Mar 25 at 8:48

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

add a comment | 

1

$begingroup$

This question already has an answer here:

• 
  What is the right first step to take when integrating $1/(2+ sin x + cos x)$? [duplicate]
  
  2 answers
  
  


• 
  Evaluating $int P(sin x, cos x) textdx$
  
  3 answers
  
  

$$int frac12+sin x+cos xdx.$$

My attempts:

1. Let $y = sin x+cos x.implies fracdydx=cos x-sin x=y'.$
   

$$int frac12+sin x+cos xdx=intfrac12+yfracdyy'.$$

And I tried to use the fact $(ln x)'=1/x,$ but $(ln(2+y))'=fracy'2+y$ : the form doesn't match. So I think I've failed at this moment.

2. Let $u=sin x.implies du=cos x dx=sqrt1-u^2dx.$

$$int frac12+sin x+cos xdx=intfrac12+u+sqrt1-u^2fracdusqrt1-u^2.$$

And it looks more uncomputable.

Both of my attempts are at a dead end. How to evaluate this integral? Any help would be appreciated.

calculus integration indefinite-integrals

share|cite|improve this question

asked Mar 25 at 7:51

user642721user642721

735

$endgroup$

marked as duplicate by Martin R, Robert Z calculus
Users with the  calculus badge can single-handedly close calculus questions as duplicates and reopen them as needed.

StackExchange.ready(function()
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$('.dupe-hammer-message-hover:not(.hover-bound)').each(function()
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);
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Mar 25 at 8:48

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

add a comment | 

$begingroup$

This question already has an answer here:

• 
  What is the right first step to take when integrating $1/(2+ sin x + cos x)$? [duplicate]
  
  2 answers
  
  


• 
  Evaluating $int P(sin x, cos x) textdx$
  
  3 answers
  
  

$$int frac12+sin x+cos xdx.$$

My attempts:

1. Let $y = sin x+cos x.implies fracdydx=cos x-sin x=y'.$
   

$$int frac12+sin x+cos xdx=intfrac12+yfracdyy'.$$

And I tried to use the fact $(ln x)'=1/x,$ but $(ln(2+y))'=fracy'2+y$ : the form doesn't match. So I think I've failed at this moment.

2. Let $u=sin x.implies du=cos x dx=sqrt1-u^2dx.$

$$int frac12+sin x+cos xdx=intfrac12+u+sqrt1-u^2fracdusqrt1-u^2.$$

And it looks more uncomputable.

Both of my attempts are at a dead end. How to evaluate this integral? Any help would be appreciated.

calculus integration indefinite-integrals

share|cite|improve this question

asked Mar 25 at 7:51

user642721user642721

735

$endgroup$

share|cite|improve this question

asked Mar 25 at 7:51

user642721user642721

735

share|cite|improve this question

asked Mar 25 at 7:51

user642721user642721

735

asked Mar 25 at 7:51

user642721user642721

735

asked Mar 25 at 7:51

user642721user642721

735

2 Answers
2

active

oldest

votes

2

$begingroup$

Hint: Use the so-called Weierstrass substitution
$$sin(x)=frac2t1+t^2$$
$$cos(x)=frac1-t^21+t^2$$
$$dx=frac21+t^2dt$$
A possible solution is given by $$sqrt 2arctan left( 1/4, left( 2,tan left( x/2 right) +2
right) sqrt 2 right)
+C$$

share|cite|improve this answer

edited Mar 25 at 8:14

answered Mar 25 at 7:54

Dr. Sonnhard GraubnerDr. Sonnhard Graubner

79k42867

$endgroup$

add a comment | 

1

$begingroup$

Hint:

$$textLet beginbmatrixsin x \ cos x \ mathrm dxendbmatrix=beginbmatrixdfrac2t1+t^2\ dfrac1-t^21+t^2\ dfrac2mathrm dt1+t^2endbmatrix$$

This transforms $R(sin x, cos x)$ to a rational function in $t$ and you can proceed with Partial Fraction Decomposition.

share|cite|improve this answer

answered Mar 25 at 7:55

Paras KhoslaParas Khosla

3,278627

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  This is not a solution, it should be posted as a comment instead (flagged).
  $endgroup$
  – user619699
  Mar 25 at 8:09
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  That's exactly my point. Hints are meant to be posted as comments.
  $endgroup$
  – user619699
  Mar 25 at 8:16
  

  
  
  
  

add a comment | 

2

$begingroup$

Hint: Use the so-called Weierstrass substitution
$$sin(x)=frac2t1+t^2$$
$$cos(x)=frac1-t^21+t^2$$
$$dx=frac21+t^2dt$$
A possible solution is given by $$sqrt 2arctan left( 1/4, left( 2,tan left( x/2 right) +2
right) sqrt 2 right)
+C$$

share|cite|improve this answer

edited Mar 25 at 8:14

answered Mar 25 at 7:54

Dr. Sonnhard GraubnerDr. Sonnhard Graubner

79k42867

$endgroup$

add a comment | 

2

$begingroup$

Hint: Use the so-called Weierstrass substitution
$$sin(x)=frac2t1+t^2$$
$$cos(x)=frac1-t^21+t^2$$
$$dx=frac21+t^2dt$$
A possible solution is given by $$sqrt 2arctan left( 1/4, left( 2,tan left( x/2 right) +2
right) sqrt 2 right)
+C$$

share|cite|improve this answer

edited Mar 25 at 8:14

answered Mar 25 at 7:54

Dr. Sonnhard GraubnerDr. Sonnhard Graubner

79k42867

$endgroup$

add a comment | 

$begingroup$

Hint: Use the so-called Weierstrass substitution
$$sin(x)=frac2t1+t^2$$
$$cos(x)=frac1-t^21+t^2$$
$$dx=frac21+t^2dt$$
A possible solution is given by $$sqrt 2arctan left( 1/4, left( 2,tan left( x/2 right) +2
right) sqrt 2 right)
+C$$

share|cite|improve this answer

edited Mar 25 at 8:14

answered Mar 25 at 7:54

Dr. Sonnhard GraubnerDr. Sonnhard Graubner

79k42867

$endgroup$

share|cite|improve this answer

edited Mar 25 at 8:14

answered Mar 25 at 7:54

Dr. Sonnhard GraubnerDr. Sonnhard Graubner

79k42867

answered Mar 25 at 7:54

Dr. Sonnhard GraubnerDr. Sonnhard Graubner

79k42867

answered Mar 25 at 7:54

Dr. Sonnhard GraubnerDr. Sonnhard Graubner

79k42867

1

$begingroup$

Hint:

$$textLet beginbmatrixsin x \ cos x \ mathrm dxendbmatrix=beginbmatrixdfrac2t1+t^2\ dfrac1-t^21+t^2\ dfrac2mathrm dt1+t^2endbmatrix$$

This transforms $R(sin x, cos x)$ to a rational function in $t$ and you can proceed with Partial Fraction Decomposition.

share|cite|improve this answer

answered Mar 25 at 7:55

Paras KhoslaParas Khosla

3,278627

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  This is not a solution, it should be posted as a comment instead (flagged).
  $endgroup$
  – user619699
  Mar 25 at 8:09
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  That's exactly my point. Hints are meant to be posted as comments.
  $endgroup$
  – user619699
  Mar 25 at 8:16
  

  
  
  
  

add a comment | 

1

$begingroup$

Hint:

$$textLet beginbmatrixsin x \ cos x \ mathrm dxendbmatrix=beginbmatrixdfrac2t1+t^2\ dfrac1-t^21+t^2\ dfrac2mathrm dt1+t^2endbmatrix$$

This transforms $R(sin x, cos x)$ to a rational function in $t$ and you can proceed with Partial Fraction Decomposition.

share|cite|improve this answer

answered Mar 25 at 7:55

Paras KhoslaParas Khosla

3,278627

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  This is not a solution, it should be posted as a comment instead (flagged).
  $endgroup$
  – user619699
  Mar 25 at 8:09
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  That's exactly my point. Hints are meant to be posted as comments.
  $endgroup$
  – user619699
  Mar 25 at 8:16
  

  
  
  
  

add a comment | 

$begingroup$

Hint:

$$textLet beginbmatrixsin x \ cos x \ mathrm dxendbmatrix=beginbmatrixdfrac2t1+t^2\ dfrac1-t^21+t^2\ dfrac2mathrm dt1+t^2endbmatrix$$

This transforms $R(sin x, cos x)$ to a rational function in $t$ and you can proceed with Partial Fraction Decomposition.

share|cite|improve this answer

answered Mar 25 at 7:55

Paras KhoslaParas Khosla

3,278627

$endgroup$

share|cite|improve this answer

answered Mar 25 at 7:55

Paras KhoslaParas Khosla

3,278627

answered Mar 25 at 7:55

Paras KhoslaParas Khosla

3,278627

answered Mar 25 at 7:55

Paras KhoslaParas Khosla

3,278627