Mixing
Proving the answer to the integral of $sin(π/x^2)$
$begingroup$
When I integrate $sin(π/x^2)$ on W|A, I get:
begin{equation}
int sinleft(frac{π}{x^2} right):dx = xsinleft(frac{pi}{x^2}right) - sqrt{2}pi Cleft( frac{sqrt{2}}{x} right) + D
end{equation}
Where $D$ is the constant of integration and $C(x)$ is the Fresnel C Integral.
But I have no idea how to prove this! There is no general integration formula for composite formulas, and the answer on W|A needs proof (there is no step-by-step and to me, it looks kind of shady. http://mathworld.wolfram.com/FresnelIntegrals.html is about the Fresnel C integral. I have found no resources anywhere and I need help with the proof!
Can anyone provide some starting points / tips?
calculus integration proof-writing indefinite-integrals
asked Jan 14 at 3:59
Math BobMath Bob
339
$endgroup$
|
show 2 more comments
$begingroup$
When I integrate $sin(π/x^2)$ on W|A, I get:
begin{equation}
int sinleft(frac{π}{x^2} right):dx = xsinleft(frac{pi}{x^2}right) - sqrt{2}pi Cleft( frac{sqrt{2}}{x} right) + D
end{equation}
Where $D$ is the constant of integration and $C(x)$ is the Fresnel C Integral.
But I have no idea how to prove this! There is no general integration formula for composite formulas, and the answer on W|A needs proof (there is no step-by-step and to me, it looks kind of shady. http://mathworld.wolfram.com/FresnelIntegrals.html is about the Fresnel C integral. I have found no resources anywhere and I need help with the proof!
Can anyone provide some starting points / tips?
calculus integration proof-writing indefinite-integrals
asked Jan 14 at 3:59
Math BobMath Bob
339
$endgroup$
1
$begingroup$
What have you attempted so far? Can you please post up. Note this is a requirement in posting on the site. If you have no idea how to start and are looking for some tips to get you going then you must state so in your question. Welcome to MSE btw!
$endgroup$
– DavidG
Jan 14 at 5:52
$begingroup$
I also seem to be getting a different answer to yours: wolframalpha.com/input/?i=integrate+cos(180%2Fx%5E2)
$endgroup$
– DavidG
Jan 14 at 5:54
$begingroup$
HINT: Integration by Parts.
$endgroup$
– DavidG
Jan 14 at 6:00
$begingroup$
Put $frac1x^2=y$ in the integral and then use by parts with $sin(y)$ as the first part and the $frac1{2cdot sqrt{y}}$ as the second part
$endgroup$
– Sauhard Sharma
Jan 14 at 7:31
2
$begingroup$
Fresnel Integral gives you a hint what you are looking for. You want $sin t^2$, so introduce $t=1/x$ and proceed with integration by parts, if needed. I'm also concerned what is the meaning of this 180 constant - if you mean degrees (judging by π on the right side, but no ° symbol is present), that is a very bad practice, degrees only make sense for angles, and as soon as you are doing analysis and have the variable in and outside the trig functions, degrees just lead to misunderstandings. Of course, °=π/180 is just a numerical constant to multiply with.
$endgroup$
– orion
Jan 14 at 10:17
|
show 2 more comments
$begingroup$
When I integrate $sin(π/x^2)$ on W|A, I get:
begin{equation}
int sinleft(frac{π}{x^2} right):dx = xsinleft(frac{pi}{x^2}right) - sqrt{2}pi Cleft( frac{sqrt{2}}{x} right) + D
end{equation}
Where $D$ is the constant of integration and $C(x)$ is the Fresnel C Integral.
But I have no idea how to prove this! There is no general integration formula for composite formulas, and the answer on W|A needs proof (there is no step-by-step and to me, it looks kind of shady. http://mathworld.wolfram.com/FresnelIntegrals.html is about the Fresnel C integral. I have found no resources anywhere and I need help with the proof!
Can anyone provide some starting points / tips?
calculus integration proof-writing indefinite-integrals
asked Jan 14 at 3:59
Math BobMath Bob
339
$endgroup$
When I integrate $sin(π/x^2)$ on W|A, I get:
begin{equation}
int sinleft(frac{π}{x^2} right):dx = xsinleft(frac{pi}{x^2}right) - sqrt{2}pi Cleft( frac{sqrt{2}}{x} right) + D
end{equation}
Where $D$ is the constant of integration and $C(x)$ is the Fresnel C Integral.
But I have no idea how to prove this! There is no general integration formula for composite formulas, and the answer on W|A needs proof (there is no step-by-step and to me, it looks kind of shady. http://mathworld.wolfram.com/FresnelIntegrals.html is about the Fresnel C integral. I have found no resources anywhere and I need help with the proof!
Can anyone provide some starting points / tips?
calculus integration proof-writing indefinite-integrals
calculus integration proof-writing indefinite-integrals
asked Jan 14 at 3:59
Math BobMath Bob
339
asked Jan 14 at 3:59
Math BobMath Bob
339
edited Jan 14 at 16:31
Math Bob
asked Jan 14 at 3:59
Math BobMath Bob
339
asked Jan 14 at 3:59
Math BobMath Bob
339
asked Jan 14 at 3:59
Math BobMath Bob
339
339
1
$begingroup$
What have you attempted so far? Can you please post up. Note this is a requirement in posting on the site. If you have no idea how to start and are looking for some tips to get you going then you must state so in your question. Welcome to MSE btw!
$endgroup$
– DavidG
Jan 14 at 5:52
$begingroup$
I also seem to be getting a different answer to yours: wolframalpha.com/input/?i=integrate+cos(180%2Fx%5E2)
$endgroup$
– DavidG
Jan 14 at 5:54
$begingroup$
HINT: Integration by Parts.
$endgroup$
– DavidG
Jan 14 at 6:00
$begingroup$
Put $frac1x^2=y$ in the integral and then use by parts with $sin(y)$ as the first part and the $frac1{2cdot sqrt{y}}$ as the second part
$endgroup$
– Sauhard Sharma
Jan 14 at 7:31
2
$begingroup$
Fresnel Integral gives you a hint what you are looking for. You want $sin t^2$, so introduce $t=1/x$ and proceed with integration by parts, if needed. I'm also concerned what is the meaning of this 180 constant - if you mean degrees (judging by π on the right side, but no ° symbol is present), that is a very bad practice, degrees only make sense for angles, and as soon as you are doing analysis and have the variable in and outside the trig functions, degrees just lead to misunderstandings. Of course, °=π/180 is just a numerical constant to multiply with.
$endgroup$
– orion
Jan 14 at 10:17
|
show 2 more comments
1
$begingroup$
What have you attempted so far? Can you please post up. Note this is a requirement in posting on the site. If you have no idea how to start and are looking for some tips to get you going then you must state so in your question. Welcome to MSE btw!
$endgroup$
– DavidG
Jan 14 at 5:52
$begingroup$
I also seem to be getting a different answer to yours: wolframalpha.com/input/?i=integrate+cos(180%2Fx%5E2)
$endgroup$
– DavidG
Jan 14 at 5:54
$begingroup$
HINT: Integration by Parts.
$endgroup$
– DavidG
Jan 14 at 6:00
$begingroup$
Put $frac1x^2=y$ in the integral and then use by parts with $sin(y)$ as the first part and the $frac1{2cdot sqrt{y}}$ as the second part
$endgroup$
– Sauhard Sharma
Jan 14 at 7:31
2
$begingroup$
Fresnel Integral gives you a hint what you are looking for. You want $sin t^2$, so introduce $t=1/x$ and proceed with integration by parts, if needed. I'm also concerned what is the meaning of this 180 constant - if you mean degrees (judging by π on the right side, but no ° symbol is present), that is a very bad practice, degrees only make sense for angles, and as soon as you are doing analysis and have the variable in and outside the trig functions, degrees just lead to misunderstandings. Of course, °=π/180 is just a numerical constant to multiply with.
$endgroup$
– orion
Jan 14 at 10:17
1
1
$begingroup$
What have you attempted so far? Can you please post up. Note this is a requirement in posting on the site. If you have no idea how to start and are looking for some tips to get you going then you must state so in your question. Welcome to MSE btw!
$endgroup$
– DavidG
Jan 14 at 5:52
$begingroup$
What have you attempted so far? Can you please post up. Note this is a requirement in posting on the site. If you have no idea how to start and are looking for some tips to get you going then you must state so in your question. Welcome to MSE btw!
$endgroup$
– DavidG
Jan 14 at 5:52
$begingroup$
I also seem to be getting a different answer to yours: wolframalpha.com/input/?i=integrate+cos(180%2Fx%5E2)
$endgroup$
– DavidG
Jan 14 at 5:54
$begingroup$
I also seem to be getting a different answer to yours: wolframalpha.com/input/?i=integrate+cos(180%2Fx%5E2)
$endgroup$
– DavidG
Jan 14 at 5:54
$begingroup$
HINT: Integration by Parts.
$endgroup$
– DavidG
Jan 14 at 6:00
$begingroup$
HINT: Integration by Parts.
$endgroup$
– DavidG
Jan 14 at 6:00
$begingroup$
Put $frac1x^2=y$ in the integral and then use by parts with $sin(y)$ as the first part and the $frac1{2cdot sqrt{y}}$ as the second part
$endgroup$
– Sauhard Sharma
Jan 14 at 7:31
$begingroup$
Put $frac1x^2=y$ in the integral and then use by parts with $sin(y)$ as the first part and the $frac1{2cdot sqrt{y}}$ as the second part
$endgroup$
– Sauhard Sharma
Jan 14 at 7:31
2
2
$begingroup$
Fresnel Integral gives you a hint what you are looking for. You want $sin t^2$, so introduce $t=1/x$ and proceed with integration by parts, if needed. I'm also concerned what is the meaning of this 180 constant - if you mean degrees (judging by π on the right side, but no ° symbol is present), that is a very bad practice, degrees only make sense for angles, and as soon as you are doing analysis and have the variable in and outside the trig functions, degrees just lead to misunderstandings. Of course, °=π/180 is just a numerical constant to multiply with.
$endgroup$
– orion
Jan 14 at 10:17
$begingroup$
Fresnel Integral gives you a hint what you are looking for. You want $sin t^2$, so introduce $t=1/x$ and proceed with integration by parts, if needed. I'm also concerned what is the meaning of this 180 constant - if you mean degrees (judging by π on the right side, but no ° symbol is present), that is a very bad practice, degrees only make sense for angles, and as soon as you are doing analysis and have the variable in and outside the trig functions, degrees just lead to misunderstandings. Of course, °=π/180 is just a numerical constant to multiply with.
$endgroup$
– orion
Jan 14 at 10:17
|
show 2 more comments
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$begingroup$
What have you attempted so far? Can you please post up. Note this is a requirement in posting on the site. If you have no idea how to start and are looking for some tips to get you going then you must state so in your question. Welcome to MSE btw!
$endgroup$
– DavidG
Jan 14 at 5:52
$begingroup$
I also seem to be getting a different answer to yours: wolframalpha.com/input/?i=integrate+cos(180%2Fx%5E2)
$endgroup$
– DavidG
Jan 14 at 5:54
$begingroup$
HINT: Integration by Parts.
$endgroup$
– DavidG
Jan 14 at 6:00
$begingroup$
Put $frac1x^2=y$ in the integral and then use by parts with $sin(y)$ as the first part and the $frac1{2cdot sqrt{y}}$ as the second part
$endgroup$
– Sauhard Sharma
Jan 14 at 7:31
2
$begingroup$
Fresnel Integral gives you a hint what you are looking for. You want $sin t^2$, so introduce $t=1/x$ and proceed with integration by parts, if needed. I'm also concerned what is the meaning of this 180 constant - if you mean degrees (judging by π on the right side, but no ° symbol is present), that is a very bad practice, degrees only make sense for angles, and as soon as you are doing analysis and have the variable in and outside the trig functions, degrees just lead to misunderstandings. Of course, °=π/180 is just a numerical constant to multiply with.
$endgroup$
– orion
Jan 14 at 10:17