Anti-derivatives and Elementary Functions

In summary, when solving problems involving integrals, it is important to note that not all functions have an anti-derivative that can be expressed in terms of elementary functions. While the Fundamental Theorem of Calculus does guarantee the existence of an anti-derivative for continuous and integrable functions, it may not be possible to find a simple expression for it. Therefore, it is not always valid to assume that an integral can be evaluated using the formula \int_a^b f(t) dt = F(b) - F(a) . This can lead to confusion and incorrect solutions, as an integral may not provide any helpful information for evaluating the anti-derivatives at the given limits.
  • #1
JG89
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I've seen this many times: someone will post a homework question asking them to prove something about an integral [tex] \int_a^b f(t) dt [/tex], where f is some arbitrary, continuous, integrable function. They will write in their proof, [tex] \int_a^b f(t) dt = F(b) - F(a) [/tex] where they're assuming F is the anti-derivative of f. They will get told that they cannot assume that f has an anti-derivative.

My question now is, if f is continuous and integrable, then doesn't it follow from the FTC that there exists a function F such that F' = f? You may not be able to express F in terms of elementary functions, but F still must exist, right? So why can't the student write in their proof [tex] \int_a^b f(t) dt = F(b) - F(a) [/tex]?
 
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  • #2
Eh are you referring to these forums (if so then okay I guess I haven't looked that closely)? I'm not sure if this entirely relates to the situation you are referring to, but I think that many people who take a first calculus course, say AP Calculus, tend to think that the equation you refer to is the definition of an integral. I think I have made this mistake before back when I didn't have a rigorous understanding of riemann integration. But yeah, if your function is continuous (and hence integrable, but we want continuity), then an antiderivative is guaranteed, but just note that F will not be any simpler than a function of the form [itex]\int_{c}^{x}f(t)\,dt,[/itex] if there is no way of expressing the antiderivative in terms of elementary functions.
 
  • #3
JG89 said:
They will get told that they cannot assume that f has an anti-derivative.

I think what is usually said is that f has no elementary anti-derivative.

Usually, the case is that OP wants to evaluate an integral exactly by using the FTC. But this doesn't work. Sure enough, the FTC holds, and

[tex]
\int_a^b f(t) dt = F(b) - F(a)
[/tex]

but no progress is made in the problem, because it won't give any help as to how to evaluate F(b) and F(a).
 

1. What is an anti-derivative?

An anti-derivative is the inverse operation of a derivative. It is a function that, when differentiated, gives the original function. In other words, it is a function whose derivative is the given function.

2. How do you find the anti-derivative of a function?

To find the anti-derivative of a function, you can use the reverse power rule, where you add one to the power of the variable and divide by the new power. You can also use a table of common anti-derivatives or integration techniques such as integration by parts or substitution.

3. What are some common elementary functions?

Some common elementary functions include polynomials, exponential functions, logarithmic functions, trigonometric functions, and their inverses. These functions are used frequently in calculus and can be differentiated and integrated easily.

4. How are anti-derivatives and definite integrals related?

An anti-derivative is the general solution to a differential equation, while a definite integral is the specific solution for a given interval. The fundamental theorem of calculus states that the definite integral of a function can be calculated by finding its anti-derivative and evaluating it at the upper and lower limits of integration.

5. Can all functions have an anti-derivative?

No, not all functions have an anti-derivative. A function must be continuous on its entire domain to have an anti-derivative. If a function has a discontinuity or an infinite number of discontinuities, it does not have an anti-derivative. Additionally, some functions, such as the Dirac delta function, do not have an anti-derivative in the traditional sense.

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