Introduction to Indefinite Integrals

Indefinite Integrals Lecture 1

Topic: Indefinite Integrals. Introduction. Definition.

Summary

In this lecture we introduce the concept of indefinite integrals. To understand this Lecture it is ESSENTIAL to know and understand what derivatives are (the ability to compute them from formulas alone is not enough) – for example, my first two lectures on derivatives on this blog will be sufficient.

What was a derivative? What will an indefinite integral be?

We arrived at the concept of the derivative in Lecture 1 on Derivatives as follows:

1. By measuring distances traveled by a sled with a stopwatch, we determined the distance function depending on time (at that time we obtained: )
2. By taking more and more precise measurements of average velocity, we calculated the exact velocity of the sled at the 2nd second of motion
3. We concluded that the method used to determine the velocity at the 2nd second could be applied to any other second of motion to determine the corresponding velocity, and in this way we arrived at the concept of the derivative of a function – that is, a function assigning to each second of motion the value of the sled’s velocity at that second

In short: given the function describing distance depending on time, we determined the function describing velocity depending on time.

It is not hard to imagine that often we must work the OTHER WAY AROUND: given the velocity function, we need to determine the distance function depending on time. In our sled example, we could imagine sitting on the sled and writing down the velocity readings from a speedometer (I’m not sure whether sleds with speedometers exist, but surely someone will invent them one day). Knowing how velocity changes with time, we would ask how distance changes with time.

Determining such a function would be precisely integration (strict definitions are coming in a moment).

You can see that the problem is not artificial at all – often we know the velocity but not the distance, and it’s not always about mechanical velocity.

What was a derivative in another interpretation? What will an indefinite integral be?

I also introduced the concept of the derivative in Lecture 2 on Derivatives without referring to velocity, but by:

1. Defining the tangent line to a graph at a point (as the limiting position of secant lines)
2. Defining the derivative at a point as the tangent of the angle of inclination of this tangent line (with respect to the OX axis)
3. Defining the derivative as the function consisting of all these tangent values from point 2 (at each x there was a tangent line and some slope)

In short: we were given a graph of a function and determined its tangent lines at every point (the tangents of the slopes of these lines were the derivative values).

We can already guess what the indefinite integral will be in this case, right? The reverse process. Given the tangent lines to a graph, we must determine the graph.

Definition of the Indefinite Integral

Definition of Antiderivative and Integration
A function $F(x)$F(x) is called an antiderivative (primitive) of the function $f(x)$f(x) on some interval if on that interval:
$F'(x) = f(x)$F′(x)=f(x)
Finding all indefinite integrals (antiderivatives) is called integration.

Examples:

The function $F(x) = x$ is an antiderivative of the function $f(x) = 1$, because $(x)’ = 1$

The function $F(x) = x^2$ is an antiderivative of the function $f(x) = 2x$, because $(x^2)’ = 2x$

The function $F(x) = e^x$ is an antiderivative of the function $f(x) = e^x$, because $(e^x)’ = e^x$

The function $F(x) = x^2 + 1$ is an antiderivative of the function $f(x) = 2x$, because $(x^2 + 1)’ = 2x$

The function $F(x) = e^x – 17$ is an antiderivative of the function $f(x) = e^x$, because $(e^x – 17)’ = e^x$

Notice that, for example, antiderivatives of the function include the functions: , as well as: , , or: (because their derivatives are always equal to ).

In this way we arrive at the theorem:

Theorem

If on some interval $F(x)$F(x) is an antiderivative of the function $f(x)$f(x), then the function $F(x) + C$F(x)+C, where $C$C is any constant, is also an antiderivative of $f(x)$f(x).

Every antiderivative of the function $f(x)$f(x) can be written in the form $F(x) + C$F(x)+C.

Proof

The proof of the first part of the theorem is simple. From the rules of differentiation we know that:$$[F(x)+C]’ = F'(x).$$

From the assumption (that $F(x)$F(x) is an antiderivative of the function $f(x)$f(x)) we know that:$$F'(x) = f(x)$$

Therefore:$$[F(x)+C]’ = F'(x) = f(x)$$

This proves that $F(x)+C$F(x)+C is an antiderivative of the function $f(x)$f(x).

As for the second part of the theorem: take any antiderivative of the function $f(x)$f(x), different from $F(x)$. Let us denote it by
$G(x)$.

Since both $F(x)$F(x) and $G(x)$G(x) are antiderivatives of $f(x)$f(x), we have:$$F'(x) = f(x)$$$$G'(x) = f(x)$$

So they have the same derivative. Functions that have the same derivative differ only by a constant (this follows from earlier theorems in analysis – sorry, I don’t have them on the blog yet – note by Krystian Karczyński), therefore:$$G(x) = F(x) + C$$G(x)=F(x)+C

Which was to be proved.

The Theorem implies that any antiderivatives differ only by a constant, so the entire family of antiderivatives can be written as:

, where is any particular antiderivative.

Naturally, this leads us to today’s main point, which is…

Definition of the Indefinite Integral

The indefinite integral is defined as the family of antiderivatives of the function $f(x)$f(x), written in the form $F(x) + C$. It is denoted by:$$\int f(x)\,dx = F(x) + C$$

The symbol: is historically very old – treat it simply as the notation for integration (just like the symbol: denotes differentiation).
The symbol: in the integral formally denotes a differential, but since I haven’t said much about differentials in these lectures yet, treat it simply as part of the integral notation.

Remark 1

Notice that from the definition of an antiderivative (a function on a certain interval) it follows that we will always compute integrals of functions defined on intervals of x, not – as sometimes in derivatives – their values at specific points using limits. Of course, an integral as a function has a value at a point, but the order is always:

1. We compute the integral of a function and obtain another function
2. We compute the value of that function at a specific point

…which is in a sense the reverse of what happened with derivatives.

Remark 2

The constant C in the indefinite integral makes sense in both interpretations of the derivative (derivative as velocity at a point and as slope of the tangent line). Indeed, consider:

1. In the first interpretation, imagine we know exactly how the sled’s velocity changed over time. Based on that, we can reconstruct how distance changed over time and determine when the sled traveled 10 meters or 100 meters. However, we CANNOT determine where the sled started its motion – halfway down the hill, at the top of the hill, or in New York. These different initial positions differ precisely by the constant C – for example, measured in meters.
2. In the second interpretation, imagine that all tangent lines to a curve are drawn on the graph. Would the angles of inclination of these tangents (which determine the derivative) change if we shifted the entire graph 4 units up or several units down? Of course not. All such graphs, identical in shape but shifted vertically, are precisely antiderivatives differing by a constant C.

While writing this post, I used:

1. “Differential and Integral Calculus. Vol. II.” G.M. Fichtenholz. 1966 edition.

END

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