Is this not a well-defined integral?
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If we consider: $int_a^bxdx$, where $a=sinx$, $b=cosx$.
How can we evaluate this particular integral, if $a$ and $b$ are both functions of $x$, which is the variable with respect to which we are integrating?
calculus
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up vote
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If we consider: $int_a^bxdx$, where $a=sinx$, $b=cosx$.
How can we evaluate this particular integral, if $a$ and $b$ are both functions of $x$, which is the variable with respect to which we are integrating?
calculus
New contributor
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
If we consider: $int_a^bxdx$, where $a=sinx$, $b=cosx$.
How can we evaluate this particular integral, if $a$ and $b$ are both functions of $x$, which is the variable with respect to which we are integrating?
calculus
New contributor
If we consider: $int_a^bxdx$, where $a=sinx$, $b=cosx$.
How can we evaluate this particular integral, if $a$ and $b$ are both functions of $x$, which is the variable with respect to which we are integrating?
calculus
calculus
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New contributor
New contributor
asked 40 mins ago
Julia Kim
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262
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4 Answers
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The variable inside the integral is a "dummy" in that it could be replaced by any other symbol. I think you could interpret $$int_sin x^cos x x , dx$$ as a sloppy way of writing, but having the same meaning as, $$int_sin x^cos x y , dy.$$
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Here is another interpretation.
Formally if $bgeq a$ then $int_a^bxdx$ is a notation for $intmathbf1_(a,b](x)xdx$.
Applying that here leads to: $$int_sin x^cos xxdx=intmathbf1_(sin x,cos x](x)xdx$$
I do not dare to say that this is the correct way of interpreting, but it illustrates at least that your question is a good question.
Personally I go for the interpretation of Umberto.
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Alternatively to Umberto P. you could also view it as:
$int_A f(x) dx$ where $A$ is a set, for example an area or volume, that is further distorted by your function $f(x)$.
In your case you would simply plug in your constraints: $A = a in mathbbR $ and for $f(x)$ you have of course: $f(x) = x$
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Doesn't matter. Just your final solution will also be a function of x.
Proceed the usual way: find the indefinite integral $$int xdx=fracx^22$$
put limits: $a=sin(x)$ and $b=cos(x)$
$$=fraccos^2x2-fracsin^2x2$$
$$=fraccos(2x)2$$
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
6
down vote
The variable inside the integral is a "dummy" in that it could be replaced by any other symbol. I think you could interpret $$int_sin x^cos x x , dx$$ as a sloppy way of writing, but having the same meaning as, $$int_sin x^cos x y , dy.$$
add a comment |Â
up vote
6
down vote
The variable inside the integral is a "dummy" in that it could be replaced by any other symbol. I think you could interpret $$int_sin x^cos x x , dx$$ as a sloppy way of writing, but having the same meaning as, $$int_sin x^cos x y , dy.$$
add a comment |Â
up vote
6
down vote
up vote
6
down vote
The variable inside the integral is a "dummy" in that it could be replaced by any other symbol. I think you could interpret $$int_sin x^cos x x , dx$$ as a sloppy way of writing, but having the same meaning as, $$int_sin x^cos x y , dy.$$
The variable inside the integral is a "dummy" in that it could be replaced by any other symbol. I think you could interpret $$int_sin x^cos x x , dx$$ as a sloppy way of writing, but having the same meaning as, $$int_sin x^cos x y , dy.$$
answered 38 mins ago
Umberto P.
36.5k12861
36.5k12861
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up vote
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Here is another interpretation.
Formally if $bgeq a$ then $int_a^bxdx$ is a notation for $intmathbf1_(a,b](x)xdx$.
Applying that here leads to: $$int_sin x^cos xxdx=intmathbf1_(sin x,cos x](x)xdx$$
I do not dare to say that this is the correct way of interpreting, but it illustrates at least that your question is a good question.
Personally I go for the interpretation of Umberto.
add a comment |Â
up vote
1
down vote
Here is another interpretation.
Formally if $bgeq a$ then $int_a^bxdx$ is a notation for $intmathbf1_(a,b](x)xdx$.
Applying that here leads to: $$int_sin x^cos xxdx=intmathbf1_(sin x,cos x](x)xdx$$
I do not dare to say that this is the correct way of interpreting, but it illustrates at least that your question is a good question.
Personally I go for the interpretation of Umberto.
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Here is another interpretation.
Formally if $bgeq a$ then $int_a^bxdx$ is a notation for $intmathbf1_(a,b](x)xdx$.
Applying that here leads to: $$int_sin x^cos xxdx=intmathbf1_(sin x,cos x](x)xdx$$
I do not dare to say that this is the correct way of interpreting, but it illustrates at least that your question is a good question.
Personally I go for the interpretation of Umberto.
Here is another interpretation.
Formally if $bgeq a$ then $int_a^bxdx$ is a notation for $intmathbf1_(a,b](x)xdx$.
Applying that here leads to: $$int_sin x^cos xxdx=intmathbf1_(sin x,cos x](x)xdx$$
I do not dare to say that this is the correct way of interpreting, but it illustrates at least that your question is a good question.
Personally I go for the interpretation of Umberto.
edited 25 mins ago
answered 33 mins ago
drhab
92.1k542124
92.1k542124
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up vote
0
down vote
Alternatively to Umberto P. you could also view it as:
$int_A f(x) dx$ where $A$ is a set, for example an area or volume, that is further distorted by your function $f(x)$.
In your case you would simply plug in your constraints: $A = a in mathbbR $ and for $f(x)$ you have of course: $f(x) = x$
add a comment |Â
up vote
0
down vote
Alternatively to Umberto P. you could also view it as:
$int_A f(x) dx$ where $A$ is a set, for example an area or volume, that is further distorted by your function $f(x)$.
In your case you would simply plug in your constraints: $A = a in mathbbR $ and for $f(x)$ you have of course: $f(x) = x$
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Alternatively to Umberto P. you could also view it as:
$int_A f(x) dx$ where $A$ is a set, for example an area or volume, that is further distorted by your function $f(x)$.
In your case you would simply plug in your constraints: $A = a in mathbbR $ and for $f(x)$ you have of course: $f(x) = x$
Alternatively to Umberto P. you could also view it as:
$int_A f(x) dx$ where $A$ is a set, for example an area or volume, that is further distorted by your function $f(x)$.
In your case you would simply plug in your constraints: $A = a in mathbbR $ and for $f(x)$ you have of course: $f(x) = x$
answered 30 mins ago
Imago
1,2211919
1,2211919
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up vote
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down vote
Doesn't matter. Just your final solution will also be a function of x.
Proceed the usual way: find the indefinite integral $$int xdx=fracx^22$$
put limits: $a=sin(x)$ and $b=cos(x)$
$$=fraccos^2x2-fracsin^2x2$$
$$=fraccos(2x)2$$
add a comment |Â
up vote
0
down vote
Doesn't matter. Just your final solution will also be a function of x.
Proceed the usual way: find the indefinite integral $$int xdx=fracx^22$$
put limits: $a=sin(x)$ and $b=cos(x)$
$$=fraccos^2x2-fracsin^2x2$$
$$=fraccos(2x)2$$
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Doesn't matter. Just your final solution will also be a function of x.
Proceed the usual way: find the indefinite integral $$int xdx=fracx^22$$
put limits: $a=sin(x)$ and $b=cos(x)$
$$=fraccos^2x2-fracsin^2x2$$
$$=fraccos(2x)2$$
Doesn't matter. Just your final solution will also be a function of x.
Proceed the usual way: find the indefinite integral $$int xdx=fracx^22$$
put limits: $a=sin(x)$ and $b=cos(x)$
$$=fraccos^2x2-fracsin^2x2$$
$$=fraccos(2x)2$$
edited 7 mins ago
answered 35 mins ago
omega
1,3472918
1,3472918
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Julia Kim is a new contributor. Be nice, and check out our Code of Conduct.
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