Introduction to Integrals: Indefinite Integral

Introduction to Integrals (Antiderivatives)

Calculus encompasses two major branches: differential calculus and integral calculus. Differentiation deals with finding the rate of change of a function. Integral calculus, in one of its foundational aspects, tackles the inverse problem: given a function $f(x)$ that represents the rate of change of some other function $F(x)$, can we find the original function $F(x)$? This inverse process is called antidifferentiation, and the function $F(x)$ we find is called an antiderivative or a primitive of $f(x)$.

The Inverse Process of Differentiation (Antidifferentiation)

Differentiation is the operation of taking a function $F(x)$ and finding its derivative, $F'(x)$. Let's denote this derivative as $f(x)$, so $F'(x) = f(x)$. Antidifferentiation is the operation of starting with the function $f(x)$ (which we know is the derivative of some function) and finding the original function $F(x)$.

Definition: A function $F(x)$ is called an antiderivative (or a primitive function) of a function $f(x)$ on an interval $I$ if the derivative of $F(x)$ with respect to $x$ is equal to $f(x)$ for all values of $x$ within that interval $I$.

The process of finding antiderivatives is also known as integration, specifically indefinite integration, which will be formalized with notation later.

The Problem of Uniqueness: The Constant of Integration

A crucial question arises immediately when seeking an antiderivative: Is the antiderivative of a function unique? Let's explore this with a simple example.

Consider the function $f(x) = 2x$. We are looking for a function $F(x)$ such that its derivative $F'(x)$ is $2x$.

• From our knowledge of differentiation rules (specifically the Power Rule), we know that the derivative of $x^2$ is $2x$. So, $F(x) = x^2$ is indeed an antiderivative of $f(x) = 2x$.
• Now, consider a slightly different function, $G(x) = x^2 + 5$. Let's differentiate $G(x)$:

  "$\frac{d}{dx}(x^2 + 5) = \frac{d}{dx}(x^2) + \frac{d}{dx}(5) = 2x + 0 = 2x$"

  So, $G(x) = x^2 + 5$ is also an antiderivative of $f(x) = 2x$.

• Consider another function, $H(x) = x^2 - 100$. Its derivative is:

  "$\frac{d}{dx}(x^2 - 100) = \frac{d}{dx}(x^2) - \frac{d}{dx}(100) = 2x - 0 = 2x$"

  So, $H(x) = x^2 - 100$ is also an antiderivative of $f(x) = 2x$.

These examples show that if $F(x)$ is an antiderivative of $f(x)$, then adding any constant $C$ to $F(x)$ results in another function, $F(x) + C$, which is also an antiderivative of $f(x)$. This is because the derivative of any constant $C$ is always zero:

Furthermore, it can be proven that if two functions have the same derivative on an interval, they can differ by at most a constant. That is, if $F'(x) = G'(x)$ for all $x$ in an interval $I$, then $G(x) - F(x) = C$ for some constant $C$, or $G(x) = F(x) + C$.

The Antiderivative as a Family of Functions: The General Antiderivative

Due to the non-uniqueness introduced by the constant term, a function $f(x)$ does not have a single, unique antiderivative. Instead, it has an entire collection or family of antiderivatives. All functions in this family have the same derivative $f(x)$ and differ from each other only by a constant value.

If $F(x)$ is any specific or "particular" antiderivative of $f(x)$ (meaning $F'(x) = f(x)$), then the general antiderivative of $f(x)$ is the set of all functions of the form $F(x) + C$, where $C$ is an arbitrary real number constant.

This constant, $C$, is called the constant of integration. Its presence signifies that there are infinitely many functions that have $f(x)$ as their derivative. To determine a specific antiderivative (a particular value of $C$), we would need additional information, such as the value of the function at a particular point (an initial condition or boundary condition).

The process of finding this general antiderivative, or the family of all antiderivatives, is represented by the notation of the indefinite integral, which is introduced next.

Indefinite Integrals: Definition and Notation ($\int f(x) dx$)

To formally represent the process of antidifferentiation and the family of all antiderivatives of a function, we use the notation of the indefinite integral. This notation is a fundamental symbol in integral calculus.

Definition of the Indefinite Integral

The indefinite integral of a function $f(x)$ with respect to $x$ is defined as the collection (or family) of all antiderivatives of $f(x)$.

If $F(x)$ is any function such that $F'(x) = f(x)$, then the set of all antiderivatives of $f(x)$ is given by $F(x) + C$, where $C$ is an arbitrary constant. The indefinite integral represents this entire family.

Notation for the Indefinite Integral

The indefinite integral of a function $f(x)$ with respect to the variable $x$ is denoted by the symbol $\int f(x) dx$.

$\int f(x) dx$

If $F(x)$ is any particular antiderivative of $f(x)$ (meaning $F'(x) = f(x)$), then the relationship between the integral notation and the family of antiderivatives is written as:

$\int f(x) dx = F(x) + C$

In this notation:

• $\int$ is called the integral sign. It is an elongated 'S', originating from the sum in Riemann sums (which relate to definite integrals, a later topic).
• $f(x)$ is called the integrand. This is the function that is being integrated (or whose antiderivative is being found).
• $dx$ is the differential. It indicates the variable with respect to which the integration is being performed. In $\int f(x) dx$, the variable of integration is $x$. If it were $\int g(t) dt$, the variable would be $t$.
• $F(x)$ is any function whose derivative is $f(x)$. Finding $F(x)$ requires recognizing the function whose derivative matches the integrand.
• $C$ is the constant of integration. As discussed earlier, it represents the entire family of antiderivatives. $C$ can be any real number.

The process of evaluating an indefinite integral is called integration. It's the inverse operation of differentiation.

The fundamental relationship between differentiation and indefinite integration can be summarized as:

$\frac{d}{dx}[F(x)] = f(x) \quad \iff \quad \int f(x) dx = F(x) + C$

This means that differentiating the result of an indefinite integral gives you back the original integrand (ignoring the constant $C$), and conversely, finding a function whose derivative is the integrand gives you the result of the indefinite integral (plus the constant $C$).

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Properties of Indefinite Integrals

Indefinite integrals follow properties that are direct consequences of the corresponding rules for derivatives (namely, the Constant Multiple Rule and the Sum/Difference Rules for differentiation). These properties make it easier to integrate combinations of functions.

Let $f(x)$ and $g(x)$ be functions for which indefinite integrals exist, and let $k$ be any real number constant.

1. Constant Multiple Rule for Integration

The indefinite integral of a constant times a function is equal to the constant times the indefinite integral of the function.

$\int k f(x) dx = k \int f(x) dx$

This property holds for any constant $k$.

Reason/Verification:

Let $\int f(x) dx = F(x) + C_1$. By definition, $\frac{d}{dx}[F(x) + C_1] = f(x)$.

Consider the right side of the property: $k \int f(x) dx = k (F(x) + C_1) = k F(x) + k C_1$.

Consider the left side of the property: $\int k f(x) dx$. Let's find its derivative:

$\frac{d}{dx}\left[ k \int f(x) dx \right] = \frac{d}{dx}[k (F(x) + C_1)] = \frac{d}{dx}[k F(x) + k C_1]$

Using the Sum and Constant Rule for differentiation:

Using the Constant Multiple Rule and Constant Rule for differentiation:

Since the derivative of $k \int f(x) dx$ is $k f(x)$, by the definition of indefinite integral, $k \int f(x) dx$ represents the family of antiderivatives of $k f(x)$. Thus, $\int k f(x) dx = k F(x) + C'$, where $C'$ is an arbitrary constant. Since $kC_1$ is also an arbitrary constant, we can set $C' = kC_1$, confirming the property.

2. Sum Rule for Integration

The indefinite integral of the sum of two functions is equal to the sum of their individual indefinite integrals.

$\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$

Reason/Verification:

Let $\int f(x) dx = F(x) + C_1$ and $\int g(x) dx = G(x) + C_2$. By definition, $F'(x) = f(x)$ and $G'(x) = g(x)$.

Consider the right side of the property: $\int f(x) dx + \int g(x) dx = (F(x) + C_1) + (G(x) + C_2) = F(x) + G(x) + (C_1 + C_2)$. Since $C_1 + C_2$ is an arbitrary constant, we can denote it as $C$. So the right side is $F(x) + G(x) + C$.

Consider the integrand on the left side: $f(x) + g(x)$. Let's find the derivative of $F(x) + G(x)$:

$\frac{d}{dx}[F(x) + G(x)] = \frac{d}{dx}F(x) + \frac{d}{dx}G(x)$ (Using Sum Rule for differentiation)

Since the derivative of $F(x) + G(x)$ is $f(x) + g(x)$, by the definition of indefinite integral, $\int [f(x) + g(x)] dx = F(x) + G(x) + C$, where $C$ is an arbitrary constant. This matches the form of the right side, confirming the property.

3. Difference Rule for Integration

The indefinite integral of the difference of two functions is equal to the difference of their individual indefinite integrals.

$\int [f(x) - g(x)] dx = \int f(x) dx - \int g(x) dx$

Reason/Verification:

Similar to the Sum Rule. Let $\int f(x) dx = F(x) + C_1$ and $\int g(x) dx = G(x) + C_2$. The right side becomes $(F(x) + C_1) - (G(x) + C_2) = F(x) - G(x) + (C_1 - C_2)$, which is $F(x) - G(x) + C$. The derivative of $F(x) - G(x)$ is $\frac{d}{dx}[F(x) - G(x)] = F'(x) - G'(x) = f(x) - g(x)$. Thus, $\int [f(x) - g(x)] dx = F(x) - G(x) + C$, confirming the property.

Important Note Regarding the Constant of Integration: When using the Sum or Difference Rule, each individual integral technically introduces its own constant of integration ($C_1, C_2$, etc.). However, the sum or difference of any number of arbitrary constants is simply another arbitrary constant. Therefore, in practice, we evaluate the integrals of the individual terms without adding a constant to each one, and add a single overall constant of integration, $C$, at the very end of the entire integration process for that expression.

For example, for $\int (f(x) + g(x) - h(x)) dx$, we find the antiderivatives $F(x), G(x), H(x)$ such that $F'=f, G'=g, H'=h$. Then the integral is $F(x) + G(x) - H(x) + C$.

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Standard Formulas of Indefinite Integrals

Just as differentiation has a set of standard rules and formulas for basic functions that serve as building blocks, integration also has a corresponding set of standard formulas for indefinite integrals. These formulas are essentially the reverse of the differentiation formulas we have learned. To verify any integration formula, you can differentiate the proposed result (the antiderivative plus the constant) and check if you get back the original integrand.

In each formula below, $C$ represents the arbitrary constant of integration.

Basic Algebraic Integrals

These are derived from the Power Rule and Constant Rule for differentiation.

• Power Rule for Integration: For any real number $n$, except for $n=-1$:

  $\int x^n dx = \frac{x^{n+1}}{n+1} + C$

  Verification: $\frac{d}{dx}\left(\frac{x^{n+1}}{n+1} + C\right) = \frac{1}{n+1} \frac{d}{dx}(x^{n+1}) + 0 = \frac{1}{n+1} (n+1)x^{(n+1)-1} = x^n$.

• Integral of a Constant:

  $\int k dx = kx + C$ (where $k$ is a constant)

  Verification: $\frac{d}{dx}(kx + C) = k(1) + 0 = k$. A special case is $\int 1 dx = \int x^0 dx = \frac{x^{0+1}}{0+1} + C = x + C$.

• Integral of $\frac{1}{x}$: The Power Rule $\int x^n dx = \frac{x^{n+1}}{n+1}$ is not valid for $n=-1$. The integral of $x^{-1} = \frac{1}{x}$ is a special case related to the derivative of the logarithm function.

  We know $\frac{d}{dx}(\ln x) = \frac{1}{x}$ for $x > 0$.

  We also know $\frac{d}{dx}(\ln (-x)) = \frac{1}{-x} \cdot \frac{d}{dx}(-x) = \frac{1}{-x} \cdot (-1) = \frac{1}{x}$ for $x < 0$.

  Combining these, $\frac{d}{dx}(\ln |x|) = \frac{1}{x}$ for all $x \neq 0$. Therefore, the integral is:

  $\int \frac{1}{x} dx = \ln |x| + C$ (for $x \neq 0$)

Trigonometric Integrals

These are the antiderivatives of the basic trigonometric functions' derivatives.

• $\int \cos x dx = \sin x + C$ (Since $\frac{d}{dx}(\sin x) = \cos x$)
• $\int \sin x dx = -\cos x + C$ (Since $\frac{d}{dx}(-\cos x) = -(-\sin x) = \sin x$)
• $\int \sec^2 x dx = \tan x + C$ (Since $\frac{d}{dx}(\tan x) = \sec^2 x$)
• $\int \text{cosec}^2 x dx = -\cot x + C$ (Since $\frac{d}{dx}(-\cot x) = -(-\text{cosec}^2 x) = \text{cosec}^2 x$)
• $\int \sec x \tan x dx = \sec x + C$ (Since $\frac{d}{dx}(\sec x) = \sec x \tan x$)
• $\int \text{cosec } x \cot x dx = -\text{cosec } x + C$ (Since $\frac{d}{dx}(-\text{cosec } x) = -(-\text{cosec } x \cot x) = \text{cosec } x \cot x$)

Finding the integrals of $\tan x, \cot x, \sec x, \text{cosec } x$ themselves requires the substitution method, which is covered in a later section on integration techniques.

Exponential Integrals

These formulas come directly from the derivatives of exponential functions.

• Integral of $e^x$:

  $\int e^x dx = e^x + C$ (Since $\frac{d}{dx}(e^x) = e^x$)

• Integral of $a^x$: For any base $a > 0$ and $a \neq 1$.

  $\int a^x dx = \frac{a^x}{\ln a} + C$

  Verification: $\frac{d}{dx}\left(\frac{a^x}{\ln a} + C\right) = \frac{1}{\ln a} \frac{d}{dx}(a^x) + 0 = \frac{1}{\ln a} (a^x \ln a) = a^x$.

Integrals Leading to Inverse Trigonometric Functions

These formulas are the reverse of the derivatives of inverse trigonometric functions (specifically $\arcsin$, $\arctan$, and $\text{arcsec}$).

• $\int \frac{1}{\sqrt{1 - x^2}} dx = \arcsin x + C$ (for $-1 < x < 1$, domain of derivative matches domain of integrand)
• $\int \frac{1}{1 + x^2} dx = \arctan x + C$ (for all real $x$)
• $\int \frac{1}{x\sqrt{x^2 - 1}} dx = \text{arcsec } |x| + C$ (for $|x| > 1$. The absolute value in $\ln|x|$ and $\text{arcsec}|x|$ ensures the domain of the antiderivative is consistent with the domain of the integrand). Note that some resources give $\text{arcsec } x$ without the absolute value, depending on the assumed range of the inverse secant. $\ln|x|$ is standard for $\int 1/x dx$.

Formulas involving negatives (e.g., $\int -\frac{1}{\sqrt{1 - x^2}} dx = \arccos x + C$) are also valid but are essentially the negative of the listed primary formulas, so they are often not listed separately.

Mastering these standard formulas, along with the linearity properties (Sum, Difference, and Constant Multiple Rules), allows for the integration of a wide range of functions.

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Integration (Applied Maths)

From an applied mathematics, physics, engineering, or economics perspective, integration (specifically indefinite integration or finding antiderivatives) is primarily used as a tool for recovering a function when its rate of change is known. If we know how fast a quantity is changing with respect to another variable, integration allows us to find the formula for the quantity itself (up to a constant).

The constant of integration $C$ that arises in indefinite integration corresponds to some initial state, fixed value, or starting point of the quantity. To determine the exact function that models a specific situation, we need additional information, usually in the form of an "initial condition" or a "boundary condition".

Recovering Functions from Rates in Applications

Here are some common applications where finding the antiderivative is essential:

• Motion (Kinematics):
  • If we know the acceleration $a(t)$ as a function of time, we can find the velocity $v(t)$ by integrating the acceleration: $v(t) = \int a(t) dt$. The constant of integration represents the initial velocity (velocity at $t=0$). To find the specific velocity function, we need to know the initial velocity $v(0)$.
  • If we know the velocity $v(t)$ as a function of time, we can find the position $s(t)$ by integrating the velocity: $s(t) = \int v(t) dt$. The constant of integration represents the initial position (position at $t=0$). To find the specific position function, we need to know the initial position $s(0)$.
  • Example: If $a(t) = g$ (constant acceleration due to gravity), then $v(t) = \int g dt = gt + C_1$. If $v(0) = v_0$, then $g(0) + C_1 = v_0 \implies C_1 = v_0$. So $v(t) = gt + v_0$. Then $s(t) = \int (gt + v_0) dt = \frac{1}{2}gt^2 + v_0 t + C_2$. If $s(0) = s_0$, then $\frac{1}{2}g(0)^2 + v_0(0) + C_2 = s_0 \implies C_2 = s_0$. So $s(t) = \frac{1}{2}gt^2 + v_0 t + s_0$.
• Economics:
  • If we know the Marginal Cost $MC(x)$ as a function of units produced $x$, we can find the total cost function $C(x)$ by integrating the marginal cost: $C(x) = \int MC(x) dx$. The constant of integration here represents the fixed costs (costs incurred even when 0 units are produced), i.e., $C(0)$. Knowing the fixed costs allows us to determine the constant $C$.
  • If we know the Marginal Revenue $MR(x)$, we can find the total revenue function $R(x)$ by integrating the marginal revenue: $R(x) = \int MR(x) dx$. The constant of integration in this case is usually 0, as typically $R(0) = 0$ (no revenue from selling 0 units).
• Growth and Decay Models: If a model describes the rate of change of a quantity, say $P$, with respect to time $t$ as $\frac{dP}{dt} = f(t)$, integrating this equation gives $P(t) = \int f(t) dt$. If, for example, the rate of growth is proportional to the current quantity ($ \frac{dP}{dt} = kP $), solving this differential equation involves integration techniques that lead to exponential models like $P(t) = P_0 e^{kt}$, where $P_0$ is the initial quantity (the constant of integration determined by $P(0)$).
• Engineering and Physics: Integration is used in countless applications to find quantities from their rates of change: finding total work done from a variable force (integral of force w.r.t. distance), finding total charge from variable current (integral of current w.r.t. time), finding volume from cross-sectional areas (integral of area w.r.t. height), solving differential equations that model physical systems.

In applied settings, finding the indefinite integral provides the general solution family, and additional information specific to the problem (initial values, boundary values, constraints) is used to find the unique particular solution by determining the value of the constant of integration $C$.

Indefinite Integrals as Family of Curves (Applied Maths Geometric Interpretation)

The result of evaluating an indefinite integral $\int f(x) dx = F(x) + C$ is not a single function or a single curve, but rather an infinite set of functions, each corresponding to a different real value of the constant of integration $C$. Geometrically, this family of functions represents a family of curves that share a unique property related to their slopes.

Geometric Interpretation of the Family of Curves $y = F(x) + C$

Let $y = F(x)$ be a particular antiderivative of $f(x)$, so $F'(x) = f(x)$. The indefinite integral gives the general antiderivative $y = F(x) + C$.

• Vertical Shifts: The graphs of functions $y = F(x) + C$ for different values of $C$ are all vertically shifted versions of each other. If you have the graph of $y = F(x)$, you can obtain the graph of $y = F(x) + C_1$ by shifting $y = F(x)$ vertically by $C_1$ units (upwards if $C_1 > 0$, downwards if $C_1 < 0$). The value of $C$ determines the vertical position of the curve.
• Same Slope at Corresponding $x$-values: A key property is that for any fixed value of $x$, the slopes of the tangent lines to *all* curves in the family $y = F(x) + C$ are exactly the same. This slope is given by the derivative of $y = F(x) + C$ with respect to $x$, which is $\frac{d}{dx}(F(x) + C) = F'(x) + 0 = F'(x)$. Since $F'(x) = f(x)$, the slope of the tangent to any curve $y=F(x)+C$ at a given $x$ is equal to $f(x)$, which is the original integrand. This means that if you draw a vertical line at $x=x_0$, all the curves in the family $y=F(x)+C$ will intersect this vertical line, and the tangent lines to these curves at the intersection points will all be parallel to each other (having the same slope $f(x_0)$).

Example: Consider the integral $\int 2x dx = x^2 + C$. This represents the family of parabolas $y = x^2 + C$. The graphs for $C=0, 1, -1, 5,$ etc., are parabolas that are vertical shifts of the basic parabola $y = x^2$. For any $x$, say $x=2$, the slope of the tangent line to $y = x^2 + C$ is $\frac{dy}{dx}\bigg|_{x=2} = 2x\bigg|_{x=2} = 2(2) = 4$. All parabolas in this family have a tangent line with slope 4 at $x=2$.

From an applied perspective, the family of curves $y = F(x) + C$ represents all possible solutions to a problem where the rate of change is given by $f(x)$. The constant $C$ reflects some initial or boundary condition. For example, if $f(x)$ is a marginal cost function, $F(x)+C$ is the total cost function, and $C$ represents the fixed cost (cost at $x=0$). Each value of $C$ corresponds to a different fixed cost, but all firms with the same marginal cost function will have total cost curves that are vertical shifts of each other.

Knowing the indefinite integral gives the general form of the relationship between the variables. To pinpoint the specific curve (the particular solution) that applies to a given scenario, we need one piece of information that relates $x$ and $y$ for a particular point, which allows us to solve for the unique value of $C$.