B Asking about integral notation

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The integral notation ##\int_{a}^{x} f(t) dt## is preferred because it avoids confusion by using a different variable, t, for integration limits. Writing it as ##\int_{a}^{x} f(x) dx## implies that the variable of integration is the same as the upper limit, which can lead to misunderstandings. The notation emphasizes that t is a dummy variable, distinct from x, which is crucial for clarity in calculus. Additionally, the expression ##F(x) = \int f(x) dx## is valid, but it should not be conflated with the definite integral that has specific limits. Proper notation is essential for accurate mathematical communication.
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Why writing $$\int_{a}^{x} f(x) dx$$ is not correct?
Why should it be ##\int_{a}^{x} f(t)dt## ?

Couldn't it be like this:
Let F(x) = ##\int f(x)dx## so ##\int_{a}^{x} f(x)dx## = F(x) - F(a)

Thanks
 
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songoku said:
Summary: Why writing $$\int_{a}^{x} f(x) dx$$ is not correct?

Why should it be ##\int_{a}^{x} f(t)dt## ?

Couldn't it be like this:
Let F(x) = ##\int f(x)dx## so ##\int_{a}^{x} f(x)dx## = F(x) - F(a)

Thanks
The integral ##\displaystyle{\int_a^b f(t)\,dt}## is short for ##\displaystyle{\int_{t=a}^{t=b} f(t)\,dt}.## If you use the same letter (##b=t##) for two different meanings then you cause confusion.
 
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songoku said:
Summary: Why writing $$\int_{a}^{x} f(x) dx$$ is not correct?

Why should it be ##\int_{a}^{x} f(t)dt## ?

Couldn't it be like this:
Let F(x) = ##\int f(x)dx## so ##\int_{a}^{x} f(x)dx## = F(x) - F(a)

Thanks
Why is it not correct to write $$\sum_{k = 1}^k a_k$$
 
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Thank you very much for the explanation fresh_42 and PeroK
 
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