Definite integration and stuff

  • Thread starter mooncrater
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    Integration
In summary: It is just a variable. The second mistake is that you are trying to integrate the function f (x) over the interval [0,x]. This is not what the equation is asking for. You should be trying to integrate f (x) over the interval [0,∞).
  • #36
certainly said:
It's visible here, @Ray Vickson is it not visible to you too...
I think I am using a mobile device and you desktop.
 
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  • #37
certainly said:
It's visible here, @Ray Vickson is it not visible to you too...

Yes, it was visible to me as soon as you posted your response.
 
  • #38
Raghav Gupta said:
I think I am using a mobile device and you desktop.

I have found that on a mobile device, a long LaTeX expression might not display properly. Some of it can run off the right side of the screen, and the device does not start a new line for it. It will start a new line for pure text, but not on a typeset LaTeX/TeX expression.
 
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  • #39
Raghav Gupta said:
Isn't Leibniz rule
$$ \frac{d}{dx} \int_{g(x)}^{h(x)} f(t) = f(h(x))h'(x) - f(g(x)) g'(x) $$ ?
So it should be
$$ \frac{d}{dx} \int_0^x e^{-t} f(x-t) = e^{-x}f(0) ? $$ as second term would be zero because differentiation of a constant which is the lower limit is zero

Your fundamental mistake is to treat x as a constant when you are differentiating by x. The definite integral will depend on x not just as a limit of the integral but also in the value of the integrand.
 
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  • #40
certainly said:
No the leibniz integral rule is:-
$$\frac{\partial}{\partial x}\int_{a(x)}^{b(x)}f(x,t)\ dt=\int_{a(x)}^{b(x)}\frac{\partial f}{\partial x}\ dt+f(b(x),x)\frac{\partial b}{\partial x}-f(a(x),x)\frac{\partial a}{\partial x}$$
Problems arising for me.
Getting first term in R.H.S
$$\int_0^x e^{-t} f'(x-t)$$ second term 0 and third term also 0 as δa/δx is 0.
 
  • #41
Raghav Gupta said:
Problems arising for me.
Getting first term in R.H.S
∫x0e−tf′(x−t)
\int_0^x e^{-t} f'(x-t) second term 0 and third term also 0 as δa/δx is 0.
i'm going to call the function to be integrated ##g(x,t)## (i.e ##f## in the equation you quoted) because it might cause some confusion.
Now, ##g(x,t)=e^{-t}f(x-t)## (this is the ##f## in the original problem)
therefore ##g(x,x)\frac{\partial x}{\partial x} = e^{-x}f(0)##
and the third term vanishes not only because ##\frac{\partial a}{\partial x} = 0## but also if look closely you'll notice that the limits of integration are 0 and 0 (EDIT:- In the recursive definition of f), thus making the integral 0 as well. This is a necessary step since the integral could diverge in which case we would have to take the limit.
 
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  • #42
N42dgCt1LyHE0h68Y4bDMLYD.jpg
Attached
 
  • #43
It is incorrect to assume that ##f(0)=0##...
And if it was mentioned in the question, you should have told us so...
 
  • #44
certainly said:
It is incorrect to assume that ##f(0)=0##...
And if it was mentioned in the question, you should have told us so...
No.. it wasn't mentioned in the question.
 
  • #45
mooncrater said:

Homework Statement


The question says:
f (x)=x2+7x +∫0x(e-tf (x-t)dt.
Find f (x).
But definitely f(0) = 0
As integral from 0 to 0 is zero.
 
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  • #46
errrrr... yes, sorry for that, looks like I'm finally getting sleepy...
 
  • #47
certainly said:
errrrr... yes, sorry for that, looks like I'm finally getting sleepy...
For you :sleep:.
 
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<h2>1. What is definite integration?</h2><p>Definite integration is a mathematical concept used to find the exact area under a curve between two specific points on a graph. It involves finding the antiderivative of a function and evaluating it at the upper and lower limits of integration.</p><h2>2. How is definite integration different from indefinite integration?</h2><p>Definite integration involves finding the exact area under a curve between two specific points, while indefinite integration involves finding the general antiderivative of a function. In other words, definite integration gives a specific numerical value, while indefinite integration gives a function.</p><h2>3. What is the purpose of using definite integration?</h2><p>Definite integration is used in various fields of science, engineering, and economics to find the total value, area, volume, or other quantities that can be represented by a function. It is also used to solve problems involving rates of change and accumulation.</p><h2>4. What are the basic steps for evaluating a definite integral?</h2><p>The basic steps for evaluating a definite integral are: 1) finding the antiderivative of the function, 2) substituting the upper and lower limits of integration into the antiderivative, 3) subtracting the two values to find the difference, and 4) simplifying the result if possible.</p><h2>5. Can definite integration be used for functions with multiple variables?</h2><p>Yes, definite integration can be used for functions with multiple variables, but it requires using multiple integrals. The process is similar to evaluating a single integral, but with additional steps for each variable. This is commonly used in physics and engineering to find the volume, surface area, and other quantities in three-dimensional space.</p>

1. What is definite integration?

Definite integration is a mathematical concept used to find the exact area under a curve between two specific points on a graph. It involves finding the antiderivative of a function and evaluating it at the upper and lower limits of integration.

2. How is definite integration different from indefinite integration?

Definite integration involves finding the exact area under a curve between two specific points, while indefinite integration involves finding the general antiderivative of a function. In other words, definite integration gives a specific numerical value, while indefinite integration gives a function.

3. What is the purpose of using definite integration?

Definite integration is used in various fields of science, engineering, and economics to find the total value, area, volume, or other quantities that can be represented by a function. It is also used to solve problems involving rates of change and accumulation.

4. What are the basic steps for evaluating a definite integral?

The basic steps for evaluating a definite integral are: 1) finding the antiderivative of the function, 2) substituting the upper and lower limits of integration into the antiderivative, 3) subtracting the two values to find the difference, and 4) simplifying the result if possible.

5. Can definite integration be used for functions with multiple variables?

Yes, definite integration can be used for functions with multiple variables, but it requires using multiple integrals. The process is similar to evaluating a single integral, but with additional steps for each variable. This is commonly used in physics and engineering to find the volume, surface area, and other quantities in three-dimensional space.

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