# What is the True Definition of Indefinite Integrals?

• cepheid
In summary, the conversation discusses the true definition of an indefinite integral, which is different from the definition of a definite integral. The definition states that the indefinite integral of a function is not a function, but a set of functions. The conversation also mentions the importance of this definition in an electrical engineering course and how it relates to Laplace transforms. Daniel's definition of an indefinite integral is also mentioned and praised by others in the conversation.
cepheid
Staff Emeritus
Gold Member
Well, this is an embarrassingly elementary question, but in my lecture slides (for an electrical engineering course, not a math course) the prof suddenly springs this claim on us:

The true definition of an indefinite integral is:

$$\int{f(t)dt} \equiv \int_{-\infty}^{t} {f(\tau) d\tau}$$

Well, I've never seen this before in my calculus text, and my attempt to make sense of it doesn't go so well. From what we know already, if F(t) is an antiderviative of f(t), then the left hand side becomes:

$$\int{f(t)dt} = F(t) + C$$

In comparison, from what I know of improper integrals, the right hand side should be:

$$\int_{-\infty}^{t} {f(\tau) d\tau} = \lim_{a \rightarrow -\infty} \int_{a}^{t} {f(\tau) d\tau}$$

$$= F(t) - [\lim_{a \rightarrow -\infty} F(a)]$$

So this "definition" is only true if the limit of the function F(t) as t approaches negative infinity exists and is either constant or zero. Why should this be true in general?

Nope,the definition of the indefinite integrals (is that an oxymoron,or what??) of real valued functions over a certain domain of $\mathbb{R}$ is simply (original notation)

$$I(f(x))=:\left \{F(x)+C\left|\right\frac{dF(x)}{dx}=f(x),\frac{dC}{dx}=0\right\}$$

So the indefinite integral of a function is not a function,but a set of functions...

Daniel.

I don't know why your prof. said that, but I suspect he introduces it, because it would be important to realize in his notation for the remainder of the course.
Since it's an engineering class, I guess he can do that

The biggest difference to note between a definite and an indefinite integral is that the former is a number, while the second is a function, or family of functions (a general notation for an antiderivative).

Haha, yes, our engineering classes are quite ridiculous sometimes compared to our math and physics classes since they take so many liberties with the math! So not to worry, I'm not in such dire straits as it first may seem (for instance, I'm well aware of the difference between indefinite and definite integrals), its just that I came across this and it seemed totally off the wall. The only thing I can think of is that since he introduced it in the context of Laplace transforms (this was in the course of him trying to show us what taking the Laplace transform of the integral of a function results in), and all of the functions in the time domain that he seems to want us to worry about are zero for t less than zero! (If they aren't so by definition, then he just multiplies them by the unit step function which "steps up" at zero, so that they are).

I like Daniel's definition. I have not seen it expressed that way before, though it makes perfect sense.

Thanks everyone.

U'll have to take care and 'report anything suspicious'...

Daniel.

## What is an indefinite integral?

An indefinite integral is a mathematical concept that represents the set of all possible antiderivatives of a given function.

## How is an indefinite integral different from a definite integral?

While an indefinite integral represents a set of functions, a definite integral represents a single numerical value. Also, a definite integral has upper and lower limits of integration, while an indefinite integral does not.

## How do you find an indefinite integral?

To find an indefinite integral, you must use the reverse process of differentiation. This involves finding a function whose derivative is the given function. This function is known as the antiderivative or primitive function.

## What is the notation used for indefinite integrals?

The notation used for indefinite integrals is ∫f(x)dx, where f(x) is the function being integrated and dx represents the variable of integration. The ∫ symbol represents the integral sign.

## What is the relationship between indefinite integrals and derivatives?

The relationship between indefinite integrals and derivatives is that the derivative of an indefinite integral is the original function. In other words, indefinite integration and differentiation are inverse operations.

Replies
10
Views
1K
Replies
2
Views
803
Replies
4
Views
2K
Replies
1
Views
1K
Replies
21
Views
2K
Replies
4
Views
7K
Replies
1
Views
1K
Replies
2
Views
1K
Replies
3
Views
1K
Replies
1
Views
1K