How to integrate $intfrac1sqrtx(x-9)(x-5),dx$?
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Integrate:
$$intfrac1sqrtx(x-9)(x-5),dx$$
I did some substitutions, but it seems not to be the right path to follow. Some hints?
Noticing that $x(x-9)(x-5) =x((x-7)^2-4)$ we have:
$$intfrac1sqrtx(x-9)(x-5),dx,,= intfrac1sqrt(y+7)(y^2-4),dy,,,=intfrac1sqrt7+2cosh(t),dt,,= intfrac2sqrtz^4+7z^2+1,dz,,= intfrac2sqrtleft(z^2+frac72right)^2- frac454,dz,,=,,???
$$
Substitutions are: $y=x-7,,,;y=2cosh(t),,,;z=e^fract2$
real-analysis integration indefinite-integrals elliptic-integrals
asked Aug 19 at 6:29
Arcticmonkey
12911
add a comment |Â
up vote
7
down vote
favorite
Integrate:
$$intfrac1sqrtx(x-9)(x-5),dx$$
I did some substitutions, but it seems not to be the right path to follow. Some hints?
Noticing that $x(x-9)(x-5) =x((x-7)^2-4)$ we have:
$$intfrac1sqrtx(x-9)(x-5),dx,,= intfrac1sqrt(y+7)(y^2-4),dy,,,=intfrac1sqrt7+2cosh(t),dt,,= intfrac2sqrtz^4+7z^2+1,dz,,= intfrac2sqrtleft(z^2+frac72right)^2- frac454,dz,,=,,???
$$
Substitutions are: $y=x-7,,,;y=2cosh(t),,,;z=e^fract2$
real-analysis integration indefinite-integrals elliptic-integrals
asked Aug 19 at 6:29
Arcticmonkey
12911
4
I am afraid that this will lead to some elliptic integral.
– Claude Leibovici
Aug 19 at 6:47
@ClaudeLeibovici the strangest thing is that I have taken it from an old exam of Mathematical Analysis 1, given in the uni I am attending, and elliptic integrals are in M.A.2
– Arcticmonkey
Aug 19 at 7:07
May be a typo in the integral ? Who knows ? The result from WA is still worse that what I was thinking about.
– Claude Leibovici
Aug 19 at 7:09
2
Who wants to bet that the problem is $intfracdxsqrt(x-9)(x-5)$ ?
– Claude Leibovici
Aug 19 at 7:22
@ClaudeLeibovici A typo is possible. I'm writing an e-mail to the professor, who wrote it
– Arcticmonkey
Aug 19 at 7:24
add a comment |Â
up vote
7
down vote
favorite
up vote
7
down vote
favorite
Integrate:
$$intfrac1sqrtx(x-9)(x-5),dx$$
I did some substitutions, but it seems not to be the right path to follow. Some hints?
Noticing that $x(x-9)(x-5) =x((x-7)^2-4)$ we have:
$$intfrac1sqrtx(x-9)(x-5),dx,,= intfrac1sqrt(y+7)(y^2-4),dy,,,=intfrac1sqrt7+2cosh(t),dt,,= intfrac2sqrtz^4+7z^2+1,dz,,= intfrac2sqrtleft(z^2+frac72right)^2- frac454,dz,,=,,???
$$
Substitutions are: $y=x-7,,,;y=2cosh(t),,,;z=e^fract2$
real-analysis integration indefinite-integrals elliptic-integrals
asked Aug 19 at 6:29
Arcticmonkey
12911
Integrate:
$$intfrac1sqrtx(x-9)(x-5),dx$$
I did some substitutions, but it seems not to be the right path to follow. Some hints?
Noticing that $x(x-9)(x-5) =x((x-7)^2-4)$ we have:
$$intfrac1sqrtx(x-9)(x-5),dx,,= intfrac1sqrt(y+7)(y^2-4),dy,,,=intfrac1sqrt7+2cosh(t),dt,,= intfrac2sqrtz^4+7z^2+1,dz,,= intfrac2sqrtleft(z^2+frac72right)^2- frac454,dz,,=,,???
$$
Substitutions are: $y=x-7,,,;y=2cosh(t),,,;z=e^fract2$
real-analysis integration indefinite-integrals elliptic-integrals
asked Aug 19 at 6:29
Arcticmonkey
12911
edited Aug 19 at 7:45
asked Aug 19 at 6:29
Arcticmonkey
12911
asked Aug 19 at 6:29
Arcticmonkey
12911
asked Aug 19 at 6:29
Arcticmonkey
12911
12911
4
I am afraid that this will lead to some elliptic integral.
– Claude Leibovici
Aug 19 at 6:47
@ClaudeLeibovici the strangest thing is that I have taken it from an old exam of Mathematical Analysis 1, given in the uni I am attending, and elliptic integrals are in M.A.2
– Arcticmonkey
Aug 19 at 7:07
May be a typo in the integral ? Who knows ? The result from WA is still worse that what I was thinking about.
– Claude Leibovici
Aug 19 at 7:09
2
Who wants to bet that the problem is $intfracdxsqrt(x-9)(x-5)$ ?
– Claude Leibovici
Aug 19 at 7:22
@ClaudeLeibovici A typo is possible. I'm writing an e-mail to the professor, who wrote it
– Arcticmonkey
Aug 19 at 7:24
add a comment |Â
4
I am afraid that this will lead to some elliptic integral.
– Claude Leibovici
Aug 19 at 6:47
@ClaudeLeibovici the strangest thing is that I have taken it from an old exam of Mathematical Analysis 1, given in the uni I am attending, and elliptic integrals are in M.A.2
– Arcticmonkey
Aug 19 at 7:07
May be a typo in the integral ? Who knows ? The result from WA is still worse that what I was thinking about.
– Claude Leibovici
Aug 19 at 7:09
2
Who wants to bet that the problem is $intfracdxsqrt(x-9)(x-5)$ ?
– Claude Leibovici
Aug 19 at 7:22
@ClaudeLeibovici A typo is possible. I'm writing an e-mail to the professor, who wrote it
– Arcticmonkey
Aug 19 at 7:24
4
4
I am afraid that this will lead to some elliptic integral.
– Claude Leibovici
Aug 19 at 6:47
I am afraid that this will lead to some elliptic integral.
– Claude Leibovici
Aug 19 at 6:47
@ClaudeLeibovici the strangest thing is that I have taken it from an old exam of Mathematical Analysis 1, given in the uni I am attending, and elliptic integrals are in M.A.2
– Arcticmonkey
Aug 19 at 7:07
@ClaudeLeibovici the strangest thing is that I have taken it from an old exam of Mathematical Analysis 1, given in the uni I am attending, and elliptic integrals are in M.A.2
– Arcticmonkey
Aug 19 at 7:07
May be a typo in the integral ? Who knows ? The result from WA is still worse that what I was thinking about.
– Claude Leibovici
Aug 19 at 7:09
May be a typo in the integral ? Who knows ? The result from WA is still worse that what I was thinking about.
– Claude Leibovici
Aug 19 at 7:09
2
2
Who wants to bet that the problem is $intfracdxsqrt(x-9)(x-5)$ ?
– Claude Leibovici
Aug 19 at 7:22
Who wants to bet that the problem is $intfracdxsqrt(x-9)(x-5)$ ?
– Claude Leibovici
Aug 19 at 7:22
@ClaudeLeibovici A typo is possible. I'm writing an e-mail to the professor, who wrote it
– Arcticmonkey
Aug 19 at 7:24
@ClaudeLeibovici A typo is possible. I'm writing an e-mail to the professor, who wrote it
– Arcticmonkey
Aug 19 at 7:24
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
4
down vote
accepted
In this case, there is a cubic expression in the square root, which, unfortunately, means elliptic integral, and related, elliptic functions.
Consider the Weierstrass' elliptic function, which is the inverse of the following integral:
$$u = int_y^infty frac ds sqrt4s^3 - g_2s -g_3$$
where $g_2, g_3$ are constants. Your integral can be readily expressed in this form by a direct substitution. Then determining the actual values of $g_2, g_3$, you can calculate the fundamental parallelogram of the elliptic function. After that, you can use the formulae that express the inverse of Weierstrass' elliptic function in terms of incomplete elliptic integrals.
Or, try Wolfram Alpha.
answered Aug 19 at 6:59
Trebor
33810
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
In this case, there is a cubic expression in the square root, which, unfortunately, means elliptic integral, and related, elliptic functions.
Consider the Weierstrass' elliptic function, which is the inverse of the following integral:
$$u = int_y^infty frac ds sqrt4s^3 - g_2s -g_3$$
where $g_2, g_3$ are constants. Your integral can be readily expressed in this form by a direct substitution. Then determining the actual values of $g_2, g_3$, you can calculate the fundamental parallelogram of the elliptic function. After that, you can use the formulae that express the inverse of Weierstrass' elliptic function in terms of incomplete elliptic integrals.
Or, try Wolfram Alpha.
answered Aug 19 at 6:59
Trebor
33810
add a comment |Â
up vote
4
down vote
accepted
In this case, there is a cubic expression in the square root, which, unfortunately, means elliptic integral, and related, elliptic functions.
Consider the Weierstrass' elliptic function, which is the inverse of the following integral:
$$u = int_y^infty frac ds sqrt4s^3 - g_2s -g_3$$
where $g_2, g_3$ are constants. Your integral can be readily expressed in this form by a direct substitution. Then determining the actual values of $g_2, g_3$, you can calculate the fundamental parallelogram of the elliptic function. After that, you can use the formulae that express the inverse of Weierstrass' elliptic function in terms of incomplete elliptic integrals.
Or, try Wolfram Alpha.
answered Aug 19 at 6:59
Trebor
33810
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
In this case, there is a cubic expression in the square root, which, unfortunately, means elliptic integral, and related, elliptic functions.
Consider the Weierstrass' elliptic function, which is the inverse of the following integral:
$$u = int_y^infty frac ds sqrt4s^3 - g_2s -g_3$$
where $g_2, g_3$ are constants. Your integral can be readily expressed in this form by a direct substitution. Then determining the actual values of $g_2, g_3$, you can calculate the fundamental parallelogram of the elliptic function. After that, you can use the formulae that express the inverse of Weierstrass' elliptic function in terms of incomplete elliptic integrals.
Or, try Wolfram Alpha.
answered Aug 19 at 6:59
Trebor
33810
In this case, there is a cubic expression in the square root, which, unfortunately, means elliptic integral, and related, elliptic functions.
Consider the Weierstrass' elliptic function, which is the inverse of the following integral:
$$u = int_y^infty frac ds sqrt4s^3 - g_2s -g_3$$
where $g_2, g_3$ are constants. Your integral can be readily expressed in this form by a direct substitution. Then determining the actual values of $g_2, g_3$, you can calculate the fundamental parallelogram of the elliptic function. After that, you can use the formulae that express the inverse of Weierstrass' elliptic function in terms of incomplete elliptic integrals.
Or, try Wolfram Alpha.
answered Aug 19 at 6:59
Trebor
33810
answered Aug 19 at 6:59
Trebor
33810
answered Aug 19 at 6:59
Trebor
33810
answered Aug 19 at 6:59
Trebor
33810
33810
add a comment |Â
add a comment |Â
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4
I am afraid that this will lead to some elliptic integral.
– Claude Leibovici
Aug 19 at 6:47
@ClaudeLeibovici the strangest thing is that I have taken it from an old exam of Mathematical Analysis 1, given in the uni I am attending, and elliptic integrals are in M.A.2
– Arcticmonkey
Aug 19 at 7:07
May be a typo in the integral ? Who knows ? The result from WA is still worse that what I was thinking about.
– Claude Leibovici
Aug 19 at 7:09
2
Who wants to bet that the problem is $intfracdxsqrt(x-9)(x-5)$ ?
– Claude Leibovici
Aug 19 at 7:22
@ClaudeLeibovici A typo is possible. I'm writing an e-mail to the professor, who wrote it
– Arcticmonkey
Aug 19 at 7:24