Evaluating indefinite integral with any hint or solution
How can I evaluate this indefinite integral.
$$
intfrac{dx}{xsqrt{x^3+x+1}}
$$
any hit would be appreciated.
integration indefinite-integrals
asked Nov 26 at 17:36
Amin Qassemi
164
add a comment |
How can I evaluate this indefinite integral.
$$
intfrac{dx}{xsqrt{x^3+x+1}}
$$
any hit would be appreciated.
integration indefinite-integrals
asked Nov 26 at 17:36
Amin Qassemi
164
1
It does seem to have a nice looking closed-form. Where is this from?
– mrtaurho
Nov 26 at 17:38
2
Some sort of elliptic integral?
– Richard Martin
Nov 26 at 17:39
add a comment |
How can I evaluate this indefinite integral.
$$
intfrac{dx}{xsqrt{x^3+x+1}}
$$
any hit would be appreciated.
integration indefinite-integrals
asked Nov 26 at 17:36
Amin Qassemi
164
How can I evaluate this indefinite integral.
$$
intfrac{dx}{xsqrt{x^3+x+1}}
$$
any hit would be appreciated.
integration indefinite-integrals
integration indefinite-integrals
asked Nov 26 at 17:36
Amin Qassemi
164
asked Nov 26 at 17:36
Amin Qassemi
164
asked Nov 26 at 17:36
Amin Qassemi
164
asked Nov 26 at 17:36
Amin Qassemi
164
asked Nov 26 at 17:36
Amin Qassemi
164
164
1
It does seem to have a nice looking closed-form. Where is this from?
– mrtaurho
Nov 26 at 17:38
2
Some sort of elliptic integral?
– Richard Martin
Nov 26 at 17:39
add a comment |
1
It does seem to have a nice looking closed-form. Where is this from?
– mrtaurho
Nov 26 at 17:38
2
Some sort of elliptic integral?
– Richard Martin
Nov 26 at 17:39
1
1
It does seem to have a nice looking closed-form. Where is this from?
– mrtaurho
Nov 26 at 17:38
It does seem to have a nice looking closed-form. Where is this from?
– mrtaurho
Nov 26 at 17:38
2
2
Some sort of elliptic integral?
– Richard Martin
Nov 26 at 17:39
Some sort of elliptic integral?
– Richard Martin
Nov 26 at 17:39
add a comment |
1 Answer
1
active
oldest
votes
This "reduces" to an incomplete elliptic integral.
Somewhat more generally,
$$ int dfrac{dx}{x sqrt{(x-a)(x-b)(x-c)}} = {frac {pm 2,i}{a sqrt {a-c}}{it EllipticPi} left( {frac {sqrt {a-x}}{sqrt {a-b}}},{frac {a-b}{a}},{frac {sqrt {a-b}}{sqrt {a-c}}}
right) }$$
(using Maple's notation). In your case you want to take $a,b,c$ to be the three roots of $x^3+x+1$ (one real root approximately $-0.682327803828019$, two complex roots approximately $0.341163901914009693 pm 1.16154139999725192,i$).
answered Nov 26 at 17:56
Robert Israel
318k23207458
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
This "reduces" to an incomplete elliptic integral.
Somewhat more generally,
$$ int dfrac{dx}{x sqrt{(x-a)(x-b)(x-c)}} = {frac {pm 2,i}{a sqrt {a-c}}{it EllipticPi} left( {frac {sqrt {a-x}}{sqrt {a-b}}},{frac {a-b}{a}},{frac {sqrt {a-b}}{sqrt {a-c}}}
right) }$$
(using Maple's notation). In your case you want to take $a,b,c$ to be the three roots of $x^3+x+1$ (one real root approximately $-0.682327803828019$, two complex roots approximately $0.341163901914009693 pm 1.16154139999725192,i$).
answered Nov 26 at 17:56
Robert Israel
318k23207458
add a comment |
This "reduces" to an incomplete elliptic integral.
Somewhat more generally,
$$ int dfrac{dx}{x sqrt{(x-a)(x-b)(x-c)}} = {frac {pm 2,i}{a sqrt {a-c}}{it EllipticPi} left( {frac {sqrt {a-x}}{sqrt {a-b}}},{frac {a-b}{a}},{frac {sqrt {a-b}}{sqrt {a-c}}}
right) }$$
(using Maple's notation). In your case you want to take $a,b,c$ to be the three roots of $x^3+x+1$ (one real root approximately $-0.682327803828019$, two complex roots approximately $0.341163901914009693 pm 1.16154139999725192,i$).
answered Nov 26 at 17:56
Robert Israel
318k23207458
add a comment |
This "reduces" to an incomplete elliptic integral.
Somewhat more generally,
$$ int dfrac{dx}{x sqrt{(x-a)(x-b)(x-c)}} = {frac {pm 2,i}{a sqrt {a-c}}{it EllipticPi} left( {frac {sqrt {a-x}}{sqrt {a-b}}},{frac {a-b}{a}},{frac {sqrt {a-b}}{sqrt {a-c}}}
right) }$$
(using Maple's notation). In your case you want to take $a,b,c$ to be the three roots of $x^3+x+1$ (one real root approximately $-0.682327803828019$, two complex roots approximately $0.341163901914009693 pm 1.16154139999725192,i$).
answered Nov 26 at 17:56
Robert Israel
318k23207458
This "reduces" to an incomplete elliptic integral.
Somewhat more generally,
$$ int dfrac{dx}{x sqrt{(x-a)(x-b)(x-c)}} = {frac {pm 2,i}{a sqrt {a-c}}{it EllipticPi} left( {frac {sqrt {a-x}}{sqrt {a-b}}},{frac {a-b}{a}},{frac {sqrt {a-b}}{sqrt {a-c}}}
right) }$$
(using Maple's notation). In your case you want to take $a,b,c$ to be the three roots of $x^3+x+1$ (one real root approximately $-0.682327803828019$, two complex roots approximately $0.341163901914009693 pm 1.16154139999725192,i$).
answered Nov 26 at 17:56
Robert Israel
318k23207458
edited Nov 26 at 18:02
answered Nov 26 at 17:56
Robert Israel
318k23207458
answered Nov 26 at 17:56
Robert Israel
318k23207458
answered Nov 26 at 17:56
Robert Israel
318k23207458
318k23207458
add a comment |
add a comment |
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1
It does seem to have a nice looking closed-form. Where is this from?
– mrtaurho
Nov 26 at 17:38
2
Some sort of elliptic integral?
– Richard Martin
Nov 26 at 17:39