Is $int_{0}^{infty} frac{sin^2 x }{x^2}dx$ equal to $int_{0}^{infty} frac{sin x }{x}dx$?
I saw this in a solution to a problem and can't see why this would be the case. Tried substition and other techniques.
integration improper-integrals
asked Nov 27 at 8:36
Nuntractatuses Amável
61812
add a comment |
I saw this in a solution to a problem and can't see why this would be the case. Tried substition and other techniques.
integration improper-integrals
asked Nov 27 at 8:36
Nuntractatuses Amável
61812
It can be a simple coincidence. Does this appear as a step in a proof ? Provide more context.
– Yves Daoust
Nov 27 at 8:57
add a comment |
I saw this in a solution to a problem and can't see why this would be the case. Tried substition and other techniques.
integration improper-integrals
asked Nov 27 at 8:36
Nuntractatuses Amável
61812
I saw this in a solution to a problem and can't see why this would be the case. Tried substition and other techniques.
integration improper-integrals
integration improper-integrals
asked Nov 27 at 8:36
Nuntractatuses Amável
61812
asked Nov 27 at 8:36
Nuntractatuses Amável
61812
edited Nov 27 at 9:04
asked Nov 27 at 8:36
Nuntractatuses Amável
61812
asked Nov 27 at 8:36
Nuntractatuses Amável
61812
asked Nov 27 at 8:36
Nuntractatuses Amável
61812
61812
It can be a simple coincidence. Does this appear as a step in a proof ? Provide more context.
– Yves Daoust
Nov 27 at 8:57
add a comment |
It can be a simple coincidence. Does this appear as a step in a proof ? Provide more context.
– Yves Daoust
Nov 27 at 8:57
It can be a simple coincidence. Does this appear as a step in a proof ? Provide more context.
– Yves Daoust
Nov 27 at 8:57
It can be a simple coincidence. Does this appear as a step in a proof ? Provide more context.
– Yves Daoust
Nov 27 at 8:57
add a comment |
1 Answer
1
active
oldest
votes
$$
int_{mathbb{R}}frac{sin^{2}x}{x^{2}}dx = left[sin^{2}xleft(-frac{1}{x}right)right]^{infty}_{-infty} - int_{mathbb{R}} 2sin xcos x left(-frac{1}{x}right)dx \
=int_{mathbb{R}}frac{sin 2x}{x}dx = int_{mathbb{R}}frac{sin y}{y}dy
$$
Here we use integration by parts and the substitution $y = 2x$.
answered Nov 27 at 9:03
Seewoo Lee
6,165826
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3015503%2fis-int-0-infty-frac-sin2-x-x2dx-equal-to-int-0-infty-frac%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$$
int_{mathbb{R}}frac{sin^{2}x}{x^{2}}dx = left[sin^{2}xleft(-frac{1}{x}right)right]^{infty}_{-infty} - int_{mathbb{R}} 2sin xcos x left(-frac{1}{x}right)dx \
=int_{mathbb{R}}frac{sin 2x}{x}dx = int_{mathbb{R}}frac{sin y}{y}dy
$$
Here we use integration by parts and the substitution $y = 2x$.
answered Nov 27 at 9:03
Seewoo Lee
6,165826
add a comment |
$$
int_{mathbb{R}}frac{sin^{2}x}{x^{2}}dx = left[sin^{2}xleft(-frac{1}{x}right)right]^{infty}_{-infty} - int_{mathbb{R}} 2sin xcos x left(-frac{1}{x}right)dx \
=int_{mathbb{R}}frac{sin 2x}{x}dx = int_{mathbb{R}}frac{sin y}{y}dy
$$
Here we use integration by parts and the substitution $y = 2x$.
answered Nov 27 at 9:03
Seewoo Lee
6,165826
add a comment |
$$
int_{mathbb{R}}frac{sin^{2}x}{x^{2}}dx = left[sin^{2}xleft(-frac{1}{x}right)right]^{infty}_{-infty} - int_{mathbb{R}} 2sin xcos x left(-frac{1}{x}right)dx \
=int_{mathbb{R}}frac{sin 2x}{x}dx = int_{mathbb{R}}frac{sin y}{y}dy
$$
Here we use integration by parts and the substitution $y = 2x$.
answered Nov 27 at 9:03
Seewoo Lee
6,165826
$$
int_{mathbb{R}}frac{sin^{2}x}{x^{2}}dx = left[sin^{2}xleft(-frac{1}{x}right)right]^{infty}_{-infty} - int_{mathbb{R}} 2sin xcos x left(-frac{1}{x}right)dx \
=int_{mathbb{R}}frac{sin 2x}{x}dx = int_{mathbb{R}}frac{sin y}{y}dy
$$
Here we use integration by parts and the substitution $y = 2x$.
answered Nov 27 at 9:03
Seewoo Lee
6,165826
answered Nov 27 at 9:03
Seewoo Lee
6,165826
answered Nov 27 at 9:03
Seewoo Lee
6,165826
answered Nov 27 at 9:03
Seewoo Lee
6,165826
6,165826
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3015503%2fis-int-0-infty-frac-sin2-x-x2dx-equal-to-int-0-infty-frac%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
It can be a simple coincidence. Does this appear as a step in a proof ? Provide more context.
– Yves Daoust
Nov 27 at 8:57