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Singular Integration

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2

$begingroup$

I am trying to simplify the following integral but getting no answer. Any help in how to get the resulting function as a function of t would be much appreciated?

t - Integrate[Abs[t - s]^(-1/2)*s, s, 0, 1]

calculus-and-analysis numerical-integration

share|improve this question

edited May 19 at 18:40

Carl Woll

81.7k3105209

asked May 19 at 18:27

Mutaz Mutaz

1016

$endgroup$

add a comment | 

2

$begingroup$

I am trying to simplify the following integral but getting no answer. Any help in how to get the resulting function as a function of t would be much appreciated?

t - Integrate[Abs[t - s]^(-1/2)*s, s, 0, 1]

calculus-and-analysis numerical-integration

share|improve this question

edited May 19 at 18:40

Carl Woll

81.7k3105209

asked May 19 at 18:27

Mutaz Mutaz

1016

$endgroup$

add a comment | 

$begingroup$

I am trying to simplify the following integral but getting no answer. Any help in how to get the resulting function as a function of t would be much appreciated?

t - Integrate[Abs[t - s]^(-1/2)*s, s, 0, 1]

calculus-and-analysis numerical-integration

share|improve this question

edited May 19 at 18:40

Carl Woll

81.7k3105209

asked May 19 at 18:27

Mutaz Mutaz

1016

$endgroup$

share|improve this question

edited May 19 at 18:40

Carl Woll

81.7k3105209

asked May 19 at 18:27

Mutaz Mutaz

1016

share|improve this question

edited May 19 at 18:40

Carl Woll

81.7k3105209

asked May 19 at 18:27

Mutaz Mutaz

1016

edited May 19 at 18:40

Carl Woll

81.7k3105209

edited May 19 at 18:40

Carl Woll

81.7k3105209

asked May 19 at 18:27

Mutaz Mutaz

1016

asked May 19 at 18:27

Mutaz Mutaz

1016

1 Answer
1

active

oldest

votes

4

$begingroup$

You just need to give Integrate an assumption:

t - Integrate[Abs[t-s]^(-1/2)*s,s,0,1, Assumptions->t ∈ Reals] //TeXForm

$t-left(
beginarraycc
{ &
beginarraycc
frac23 left(2 (-t)^3/2+2 sqrt1-t t+sqrt1-tright) & tleq 0 \
-frac23 left(-2 t^3/2+2 sqrtt-1 t+sqrtt-1right) & tgeq 1 \
frac23 left(2 t^3/2+2 sqrt1-t t+sqrt1-tright) & textTrue \
endarray
\
endarray
right)$

share|improve this answer

answered May 19 at 18:39

Carl WollCarl Woll

81.7k3105209

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thank you for your comment. I just have a question, do you have an idea how to find a numerical solution for Integrate[Abs[t - s]^(-1/2)*u(s), s, 0, 1] assuming that we dont know the solution u?
  $endgroup$
  – Mutaz
  May 19 at 19:18
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Mutaz, better to ask a new question (referencing this answer) than request additional help in the comments.
  $endgroup$
  – mikado
  May 19 at 19:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ok, will do. Thank you!
  $endgroup$
  – Mutaz
  May 19 at 19:31
  

  
  
  
  

add a comment | 

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4

$begingroup$

You just need to give Integrate an assumption:

t - Integrate[Abs[t-s]^(-1/2)*s,s,0,1, Assumptions->t ∈ Reals] //TeXForm

$t-left(
beginarraycc
{ &
beginarraycc
frac23 left(2 (-t)^3/2+2 sqrt1-t t+sqrt1-tright) & tleq 0 \
-frac23 left(-2 t^3/2+2 sqrtt-1 t+sqrtt-1right) & tgeq 1 \
frac23 left(2 t^3/2+2 sqrt1-t t+sqrt1-tright) & textTrue \
endarray
\
endarray
right)$

share|improve this answer

answered May 19 at 18:39

Carl WollCarl Woll

81.7k3105209

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thank you for your comment. I just have a question, do you have an idea how to find a numerical solution for Integrate[Abs[t - s]^(-1/2)*u(s), s, 0, 1] assuming that we dont know the solution u?
  $endgroup$
  – Mutaz
  May 19 at 19:18
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Mutaz, better to ask a new question (referencing this answer) than request additional help in the comments.
  $endgroup$
  – mikado
  May 19 at 19:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ok, will do. Thank you!
  $endgroup$
  – Mutaz
  May 19 at 19:31
  

  
  
  
  

add a comment | 

4

$begingroup$

You just need to give Integrate an assumption:

t - Integrate[Abs[t-s]^(-1/2)*s,s,0,1, Assumptions->t ∈ Reals] //TeXForm

$t-left(
beginarraycc
{ &
beginarraycc
frac23 left(2 (-t)^3/2+2 sqrt1-t t+sqrt1-tright) & tleq 0 \
-frac23 left(-2 t^3/2+2 sqrtt-1 t+sqrtt-1right) & tgeq 1 \
frac23 left(2 t^3/2+2 sqrt1-t t+sqrt1-tright) & textTrue \
endarray
\
endarray
right)$

share|improve this answer

answered May 19 at 18:39

Carl WollCarl Woll

81.7k3105209

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thank you for your comment. I just have a question, do you have an idea how to find a numerical solution for Integrate[Abs[t - s]^(-1/2)*u(s), s, 0, 1] assuming that we dont know the solution u?
  $endgroup$
  – Mutaz
  May 19 at 19:18
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Mutaz, better to ask a new question (referencing this answer) than request additional help in the comments.
  $endgroup$
  – mikado
  May 19 at 19:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ok, will do. Thank you!
  $endgroup$
  – Mutaz
  May 19 at 19:31
  

  
  
  
  

add a comment | 

$begingroup$

You just need to give Integrate an assumption:

t - Integrate[Abs[t-s]^(-1/2)*s,s,0,1, Assumptions->t ∈ Reals] //TeXForm

$t-left(
beginarraycc
{ &
beginarraycc
frac23 left(2 (-t)^3/2+2 sqrt1-t t+sqrt1-tright) & tleq 0 \
-frac23 left(-2 t^3/2+2 sqrtt-1 t+sqrtt-1right) & tgeq 1 \
frac23 left(2 t^3/2+2 sqrt1-t t+sqrt1-tright) & textTrue \
endarray
\
endarray
right)$

share|improve this answer

answered May 19 at 18:39

Carl WollCarl Woll

81.7k3105209

$endgroup$

t - Integrate[Abs[t-s]^(-1/2)*s,s,0,1, Assumptions->t ∈ Reals] //TeXForm

share|improve this answer

answered May 19 at 18:39

Carl WollCarl Woll

81.7k3105209

answered May 19 at 18:39

Carl WollCarl Woll

81.7k3105209

answered May 19 at 18:39

Carl WollCarl Woll

81.7k3105209