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Ahmed Mehedi
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- Derivative of definite integral
If $$F(x)=\int_{a}^{b}f(x)dx$$ implies $$F'(x)=\int_{a}^{b}f'(x)dx$$?
Ahmed Mehedi said:Summary:: Derivative of definite integral
If $$F(x)=\int_{a}^{b}f(x)dx$$
Sorry. I understood after posting it.PeroK said:##F## is a real number. It is not a function of ##x##.
PeroK said:##F## is a real number. It is not a function of ##x##.
Ahmed Mehedi said:What I actually wants to know here is as the following:
If ##f## and ##g## are differentiable wrt ##x## and if $$\int g(x)dx=\int f(x)dx$$ can we write $$\int g'(x)dx=\int f'(x)dx$$? In short, can we pass the differentiation operator inside the integration operator without sacrifising the equality?
Ahmed Mehedi said:What I actually wants to know here is as the following:
If ##f## and ##g## are differentiable wrt ##x## and if $$\int g(x)dx=\int f(x)dx$$ can we write $$\int g'(x)dx=\int f'(x)dx$$? In short, can we pass the differentiation operator inside the integration operator without sacrifising the equality?
PeroK said:I suspect there is a lot missing regarding the context of this question. Are these definite or indefinite integrals?
Can you not look at some examples yourself?Ahmed Mehedi said:Now I wants to know about the indefinite integral. But, I would be happier if you discuss both.
PeroK said:Can you not look at some examples yourself?
What about looking at Taylor series?Ahmed Mehedi said:I guess examples may reject or accept the claim. But they don't comprise a proof of the claim. I want to know whether it can be used as an identity irrespective of any special circumstances.
PeroK said:What about looking at Taylor series?
Ahmed Mehedi said:I intuitively look at the following steps. I would be happy if you point me out my mistakes if there is any:
$$\int f(x)dx=\int g(x)dx$$
$$\frac{d}{dx}\int f(x)dx=\frac{d}{dx}\int g(x)dx$$
$$f(x)=g(x)$$
$$f'(x)=g'(x)$$
$$\int f'(x)dx=\int g'(x)dx$$
Above steps appear to serve the purpose. But I guess I have been missing something as it seems so straight forward.
PeroK said:What about:
$$\int f(x)dx=\int g(x)dx + C$$
$$\int (f(x) - g(x))dx= C$$
$$ f(x) = g(x)$$
I assumed ##f, g## are continuous.Ahmed Mehedi said:This will also serve the purpose I guess. But, both of the approaches have one serious flaw. In the first line we are assuming equality between two integrals. While in the third line we are deriving the equality between the two integrands of the 1st line. That is we are claiming that equality of the two integrals confers the equality between the two integrands which can not be true as a general case.
I guess that does not help much. g(t) can be any time-shifted version of f(t). Area under them may be equal while functions themselves are not.PeroK said:I assumed ##f, g## are continuous.
If ##f## is not continuous, how are you going to differentiate it?Ahmed Mehedi said:I guess that does not help much. g(t) can be any time-shifted version of f(t). Area under them may be equal while functions themselves are not.
Ahmed Mehedi said:Summary:: Derivative of definite integral
If $$F(x)=\int_{a}^{b}f(x)dx$$ implies $$F'(x)=\int_{a}^{b}f'(x)dx$$?
Assume ##f(x)=x^2## and ##g(x)=(x-a)^2##. Both are continuous. Area under them are equal. Yet ##f(x)\neq g(x)##PeroK said:If ##f## is not continuous, how are you going to differentiate it?
Ahmed Mehedi said:Assume ##f(x)=x^2## and ##g(x)=(x-a)^2##. Both are continuous. Area under them are equal. Yet ##f(x)\neq g(x)##
You said indefinite integral. Hence the constant of integration ##C## in my post.Ahmed Mehedi said:Now I wants to know about the indefinite integral. But, I would be happier if you discuss both.
TheDS1337 said:You are confusing an integrand with a function, notice that after integration of f(x), the dependence of x is totaly lost and you have a real number, hence F'(x) = 0
PeroK said:You said indefinite integral. Hence the constant of integration ##C## in my post.
As has already been pointed out, this is false. Here F(x) is a constant equal to the value of the definite integral on the right side of that equation. Hence, F'(x) = 0.Ahmed Mehedi said:If ##F(x)=\int_{a}^{b}f(x)dx## implies ##F'(x)=\int_{a}^{b}f'(x)dx##?
If you're talking about the area under these curves, you're necessarily talking about definite integrals. For the areas to be equal, the limits of integration would have to be different.Ahmed Mehedi said:Assume ##f(x)=x^2## and ##g(x)=(x-a)^2##. Both are continuous. Area under them are equal. Yet ##f(x)\neq g(x)##
Ahmed Mehedi said:I intuitively look at the following steps. I would be happy if you point me out my mistakes if there is any:
$$\int f(x)dx=\int g(x)dx$$
$$\frac{d}{dx}\int f(x)dx=\frac{d}{dx}\int g(x)dx$$
$$f(x)=g(x)$$
$$f'(x)=g'(x)$$
$$\int f'(x)dx=\int g'(x)dx$$
Above steps appear to serve the purpose. But I guess I have been missing something as it seems so straight forward.
Mark44 said:In post #1 you wrote this:
As has already been pointed out, this is false. Here F(x) is a constant equal to the value of the definite integral on the right side of that equation. Hence, F'(x) = 0.
If you're talking about the area under these curves, you're necessarily talking about definite integrals. For the areas to be equal, the limits of integration would have to be different.
For example ##\int_0^1 x^2 dx \ne \int_0^1 (x - a)^2 dx##, but ##\int_0^1 x^2 dx = \int_a^{1 + a}1 (x - a)^2 dx##.
You seem to be unclear on the differences between an indefinite integral, a definite integral, and an integral that represents a function.
Indefinite integral: ##\int f(x) dx## -- represents a function
Definite integral: ##\int_a^b f(x) dx## -- represesnts a number
Definite integral as a function: ##\int_a^t f(x)dx## -- represents a function of t
Delta2 said:I believe your 3 steps are correct and consistent with the definition of the indefinite integral as the anti-derivative.
However if you going to attempt to do similar steps with definite integrals for example to start from
$$\int_a^b f(x)dx=\int_a^b g(x)dx \text { (1)}$$
then if you demand that the above equality holds for any ##a,b## then you also can conclude that ##f(x)=g(x)##. We use an argument like this in classical electromagnetism to switch from the integral form of Maxwell's equations to the differential form of them.
if (1) holds only for specific## a,b ##then you simply cannot infer that ##f(x)=g(x)## and taking the derivative with respect to x of (1) you ll just end up with ##0=0##.
Delta2 said:I believe your 3 steps are correct and consistent with the definition of the indefinite integral as the anti-derivative.
However if you going to attempt to do similar steps with definite integrals for example to start from
$$\int_a^b f(x)dx=\int_a^b g(x)dx \text { (1)}$$
then if you demand that the above equality holds for any ##a,b## then you also can conclude that ##f(x)=g(x)##. We use an argument like this in classical electromagnetism to switch from the integral form of Maxwell's equations to the differential form of them.
if (1) holds only for specific## a,b ##then you simply cannot infer that ##f(x)=g(x)## and taking the derivative with respect to x of (1) you ll just end up with ##0=0##.
The derivative of a definite integral is the rate of change of the integral with respect to its upper limit. It represents the slope of the tangent line to the integral curve at a specific point.
The derivative of a definite integral can be calculated using the Fundamental Theorem of Calculus, which states that the derivative of a definite integral is equal to the integrand evaluated at the upper limit of integration.
The derivative of a definite integral is used in various fields of science and engineering to analyze rates of change and predict future behavior. It is particularly useful in physics, economics, and engineering.
Yes, the derivative of a definite integral can be negative. This indicates that the integral is decreasing at that specific point.
No, the derivative of a definite integral is not always equal to the original integrand. The Fundamental Theorem of Calculus only applies when the upper limit of integration is a constant. If the upper limit is a variable, the derivative will include an additional term involving the derivative of the upper limit.