Introduction
Calculus Guide is a book to get started with calculus. It not a reference nor a textbook but a simple guide to get overview of calculus and help you learn calculus.
Contributing
Calculus Guide is free and open source guide. You can find the source code on GitHub and you can report any issues on the GitHub issue tracker. And Contributions are most welcome you can read the CONTRIBUTING guide and consider opening a pull request.
License
The mdBook source and documentation are released under the Mozilla Public License v2.0.
TODO
- Calculus 1
- Calculus 2
- Calculus 3
Precalculus
Precalulus is studied before the study of calculus. These include topics such as functions, trigonometry, and algebraic concepts. Precalculus is designed to prepare you for the study of calculus by introducing and reinforcing important mathematical concepts and principles. Through your study of precalculus, you will gain a deeper understanding of mathematical concepts and principles, and you will be better able to apply these principles to real-world situations.
For reference
Precalculus: An Investigation of Functions (2nd Ed)
https://archive.org/details/MITRES.18-008
Calculus 1
Limits
What is Limits?
Limits in maths are defined as the values that a function approaches the output for the given input values. Limits play a vital role in calculus and mathematical analysis and are used to define integrals, derivatives, and continuity. It is used in the analysis process, and it always concerns the behavior of the function at a particular point. The limit of a sequence is further generalized in the concept of the limit of a topological net and related to the limit and direct limit in the theory category. Generally, the integrals are classified into two types namely, definite and indefinite integrals. For definite integrals, the upper limit and lower limits are defined properly. Whereas indefinite integrals are expressed without limits, and it will have an arbitrary constant while integrating the function.
Let $f\left( x \right)$ be a function defined on an interval that contains $x = a$, except possibly at $x = a$. Then we say that,
$$
\mathop {\lim }\limits_{x \to a} f\left( x \right) = L
$$
if for every number $\varepsilon > 0$ there is some number $\delta > 0$ such that
$$[\left| {f\left( x \right) - L} \right| < \varepsilon \hspace{0.5in}{\mbox{whenever}}\hspace{0.5in}0 < \left| {x - a} \right| < \delta ] $$
Where Limits is been used?
Example of Limits
$$ \lim_{x \to \pi} \sin(x/2+ \sin(x)) $$
Solving Limits with python
Limits can be solve by:
-
Substitution: Only when function is continuous
-
Factoring
-
Tabular & Approximation method: When it's by $\frac{0}{0}$
-
Conjugate
-
Formal Method
-
Infinite Limits and Rational Functions
-
L'Hôpital's Rule
Resources
Continuity
What is Continuity?
continuity is a property of a function that describes how the function behaves at different points in its domain. A function is considered continuous if it is defined for all values of its input variables and if it has a finite and well-defined output value for all of those inputs. This means that there are no "jumps" or "breaks" in the function, and that the output of the function changes smoothly as the input is varied.
Mathematically, a function is considered continuous at a given point if the following two conditions are met:
-
The function is defined at the point (i.e., it is not undefined or infinite).
-
The limit of the function as the input approaches the point is equal to the value of the function at the point. This can be written mathematically as follows:
$$ \lim_{x \to a} f(x) = f(a) $$
Where a is the point at which the function is being evaluated for continuity, and f(x) is the function itself. If this limit exists and is equal to the value of the function at the point, then the function is considered continuous at that point.
Resources
One-Sided Limits
What is One-Sided Limits?
One-sided limit is a type of limit that describes how a function behaves as the input approaches a given value from one side only. This is different from a standard limit, which describes how a function behaves as the input approaches a value from both sides.
One-sided limits are often used to describe how a function behaves at points where it is not defined or is not continuous. For example, if a function has a "jump" or a "break" at a certain point, the one-sided limits of the function at that point will be different depending on which side of the point the input is approaching from.
Mathematically, a one-sided limit of a function $f(x)$ at a point a is defined as follows:
$$ \lim_{x \to a^-} f(x) = L $$
This equation states that the one-sided limit of the function $f(x)$ at the point a from the left (i.e., as x approaches a from values less than a) is equal to the value L. Similarly, the one-sided limit of the function at the point a from the right can be written as follows:
$$ \lim_{x \to a^+} f(x) = L $$
This equation states that the one-sided limit of the function $f(x)$ at the point a from the right (i.e., as $x$ approaches a from values greater than a) is equal to the value L.
- If the one-sided limits from both sides are equal, then the function is considered continuous at the point a.
- If the one-sided limits from both sides are not equal, then the function is considered discontinuous means the Limit doesn't exists.
Resources
Limit Properties
Properties of Limits
Limit properties are a set of rules and principles that describe how the limit of a function can be calculated or estimated based on the limits of other functions. These properties allow mathematicians to manipulate and simplify expressions involving limits, and to evaluate limits that might otherwise be difficult or impossible to calculate directly.
Addition Property
The addition property of limits states that if we have two functions, f and g, that are continuous at a point a, then the limit of their sum as x approaches a is equal to the sum of their limits. In other words,
$$\lim_{x\to a} (f(x)+g(x))=\lim_{x\to a}f(x)+\lim_{x\to a}g(x)$$
This property holds true for all real numbers a, including infinity and negative infinity, as long as the limits in question exist.
To understand this concept better, consider the following example:
Suppose we want to find the limit of the function h(x) = 3x^2 + 4x - 2 as x approaches 1. We can rewrite h(x) as the sum of two functions: f(x) = 3x^2 and g(x) = 4x - 2.
Using the addition property of limits, we know that
$$\lim_{x\to 1} h(x) = \lim_{x\to 1} (f(x)+g(x)) = \lim_{x\to 1}f(x) + \lim_{x\to 1}g(x)$$
Now, we can find the limits of f(x) and g(x) individually:
$$\lim_{x\to 1}f(x) = \lim_{x\to 1} 3x^2 = 3(1)^2 = 3$$
$$\lim_{x\to 1}g(x) = \lim_{x\to 1} (4x-2) = 4(1)-2 = 2$$
Plugging these values back into the original equation, we get:
$$\lim_{x\to 1} h(x) = \lim_{x\to 1} (3x^2+4x-2) = 3 + 2 = 5$$
Therefore, the limit of h(x) as x approaches 1 is equal to 5.
In summary, the addition property of limits allows us to simplify complex functions by breaking them down into simpler parts and applying limits individually, which aids in finding the limit of the original function.
Some of the most important limit properties include the following:
- The limit of a constant is the constant itself. This means that if a function f(x) is a constant value c, then the limit of f(x) as x approaches some value a is also c. This can be written mathematically as follows:
$$ \lim_{x \to a} c = c $$
- The limit of the sum (or difference) of two functions is equal to the sum (or difference) of the limits of those functions. This means that if f(x) and g(x) are two functions, and if the limits of f(x) and g(x) as x approaches a are L and M, respectively, then the limit of their sum or difference is equal to L + M or L - M, as appropriate. This can be written mathematically as follows:
$$
\begin{aligned}
\lim_{x \to a} [f(x) + g(x)] &= \lim_{x \to a} f(x) + \lim_{x \to a} g(x)
\lim_{x \to a} [f(x) - g(x)] &= \lim_{x \to a} f(x) - \lim_{x \to a} g(x)
\end{aligned}
$$
- The limit of a product (or quotient) of two functions is equal to the product (or quotient) of the limits of those functions, provided the denominator of the quotient is not equal to zero. This means that if f(x) and g(x) are two functions, and if the limits of f(x) and g(x) as x approaches a are L and M, respectively, then the limit of their product or quotient is equal to L * M or L / M, as appropriate. This can be written mathematically as follows:
$$
\begin{aligned}
\lim_{x \to a} [f(x) * g(x)] &= \lim_{x \to a} f(x) * \lim_{x \to a} g(x)
\lim_{x \to a} \frac{f(x)}{g(x)} &= \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}
\end{aligned}
$$
These properties provide a set of rules and principles that can be used to manipulate and simplify expressions involving limits, and to evaluate limits that might otherwise be difficult or impossible to calculate directly.
Resources
Computing Limits
Computing Limits
Resources
Infinite Limits
Infinite Limits
An infinite limit is a type of limit that describes how a function behaves as the input values approach a specific point from one side or the other, but never actually reach that point. This can occur if the function has an unbounded output value as the input approaches the point, or if the function becomes undefined at the point.
In general, an infinite limit is represented by the symbol ±∞, where the sign indicates whether the limit is approached from the left or the right. For example, if a function f(x) has an infinite limit as x approaches a point a from the left, this can be written mathematically as follows:
$$ \lim_{x \to a^-} f(x) = -\infty $$
Similarly, if a function f(x) has an infinite limit as x approaches a point a from the right, this can be written mathematically as follows:
$$ \lim_{x \to a^+} f(x) = \infty $$
In both cases, the symbol ±∞ indicates that the function has an unbounded output value as the input approaches the point a, but never actually reaches it. Infinite limits are often encountered in calculus when working with functions that have vertical asymptotes or other types of singularities at certain points in their domain.
Resources
Derivatives
What is Derivatives?
The derivative is the instantaneous rate of change of a function with respect to one of its variables. This is equivalent to finding the slope of the tangent line to the function at a point. Let's use the view of derivatives as tangents to motivate a geometric definition of the derivative.
The derivative of \(f\left( x \right)\) with respect to x is the function \(f'\left( x \right)\) as, $$\begin{equation}\mathop {\lim }\limits_{h \to 0} \frac{{f\left( {a + h} \right) - f\left( a \right)}}{h} \end{equation}$$
Example of Derivatives
$$\frac{d}{dx} (\log_5 (x))^{x/2}$$
Resources
Differentiation Formulas
Differention Formulas
Differentiation formulas are mathematical relationships that allow us to find the derivative of a function in terms of other functions or variables. They are a useful tool in calculus for finding the derivative of functions that are not easily differentiated using the standard rules of calculus.
Here are some common differentiation formulas:
The power rule: The power rule states that the derivative of a function of the form $f(x) = x^n$ is given by:
$$\frac{d}{dx} x^n = nx^{n-1}$$
The constant multiple rule: The constant multiple rule states that the derivative of a function of the form $f(x) = c \cdot g(x)$ is given by:
$$\frac{d}{dx} (c \cdot g(x)) = c \cdot \frac{d}{dx} g(x)$$
where $c$ is a constant.
The sum rule: The sum rule states that the derivative of a function of the form $f(x) = g(x) + h(x)$ is given by:
$$\frac{d}{dx} (g(x) + h(x)) = \frac{d}{dx} g(x) + \frac{d}{dx} h(x)$$
The product rule: The product rule states that the derivative of a function of the form $f(x) = g(x) \cdot h(x)$ is given by:
$$\frac{d}{dx} (g(x) \cdot h(x)) = g(x) \cdot \frac{d}{dx} h(x) + h(x) \cdot \frac{d}{dx} g(x)$$
These are just a few examples of common differentiation formulas. There are many other formulas that are used to find the derivative of different types of functions, such as the quotient rule, the chain rule, and the implicit differentiation rule.
Product And Quotient Rule
Product And Quotient
The product rule and quotient rule are two important rules in calculus that allow us to find the derivative of a function that is the product or quotient of two other functions.
The product rule states that the derivative of the product of two functions is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function. This can be written mathematically as follows:
$$\frac{d}{dx}[f(x) \cdot g(x)] = f(x) \cdot \frac{dg}{dx} + g(x) \cdot \frac{df}{dx}$$
Here is an example of the product rule in action:
Suppose we have two functions f(x) and g(x) defined as follows:
$$f(x) = x^2 + 1$$
$$g(x) = x^3 + 2x$$
To find the derivative of the product f(x)g(x), we use the product rule as follows:
$$\frac{d}{dx}[f(x) \cdot g(x)] = f(x) \cdot \frac{dg}{dx} + g(x) \cdot \frac{df}{dx}$$
The derivative of f(x) with respect to x is simply 2x, and the derivative of g(x) with respect to x is 3x^2 + 2. Substituting these into the equation above, we get:
$$\frac{d}{dx}[f(x) \cdot g(x)] = (x^2 + 1) \cdot (3x^2 + 2) + (x^3 + 2x) \cdot 2x$$
Evaluating this expression, we get:
$$\frac{d}{dx}[f(x) \cdot g(x)] = 3x^4 + 6x^2 + 2x^3 + 4x + 2x^2 + 2x$$
This can be simplified to:
$$\frac{d}{dx}[f(x) \cdot g(x)] = 3x^4 + 8x^3 + 8x^2 + 6x$$
Therefore, the derivative of the product of f(x) and g(x) is 3x^4 + 8x^3 + 8x^2 + 6x.
The quotient rule states that the derivative of the quotient of two functions is equal to the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. This can be written mathematically as follows:
$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x) \cdot \frac{df}{dx} - f(x) \cdot \frac{dg}{dx}}{g(x)^2}$$
Here is an example of the quotient rule in action:
Suppose we have two functions f(x) and g(x) defined as follows:
$$f(x) = x^2 + 1$$
$$g(x) = x^3 + 2x$$
To find the derivative of the quotient f(x)/g(x), we use the quotient rule as follows:
Suppose we have two functions f(x) and g(x) defined as follows:
$$f(x) = x^2 + 1$$
$$g(x) = x^3 + 2x$$
To find the derivative of the quotient f(x)/g(x), we use the quotient rule as follows:
$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x) \cdot \frac{df}{dx} - f(x) \cdot \frac{dg}{dx}}{g(x)^2}$$
The derivative of f(x) with respect to x is simply 2x, and the derivative of g(x) with respect to x is 3x^2 + 2. Substituting these into the equation above, we get:
$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{(x^3 + 2x) \cdot 2x - (x^2 + 1) \cdot (3x^2 + 2)}{(x^3 + 2x)^2}$$
Evaluating this expression, we get:
$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{2x^4 + 4x^2 - 3x^4 - 6x^2 - 2x^2 - 2}{x^6 + 4x^4 + 4x^2}$$
This can be simplified to:
$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{-x^4 - 2x^2 + 2x^2 + 2}{x^6 + 4x^4 + 4x^2}$$
And further simplified to:
$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{-x^4 + 2x^2 + 2}{x^6 + 4x^4 + 4x^2}$$
Therefore, the derivative of the quotient of f(x) and g(x) is -x^4 + 2x^2 + 2.
Chain Rule
Chain Rule
The chain rule is a method for finding the derivative of a composite function. In other words, it allows you to find the rate of change of one function with respect to another.
The general form of the chain rule is as follows:
If y = f(u) and u = g(x), then the derivative of y with respect to x is given by:
$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$
In other words, the derivative of y with respect to x is equal to the derivative of y with respect to u, multiplied by the derivative of u with respect to x.
Here is an example of the chain rule in action:
Suppose we have a function y = f(x) defined as:
$$y = (x^2 + 1)^3$$
To find the derivative of this function, we can use the chain rule as follows:
First, we define a new variable u that is equal to x^2 + 1. This gives us:
$$u = x^2 + 1$$
Next, we define a new function y = f(u) that is equal to u^3. This gives us:
$$y = (x^2 + 1)^3 = u^3$$
Now we can use the chain rule to find the derivative of y with respect to x. We have:
$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$
The derivative of y with respect to u is simply 3u^2, so:
$$\frac{dy}{dx} = 3u^2 \cdot \frac{du}{dx}$$
To find the derivative of u with respect to x, we simply take the derivative of the original equation u = x^2 + 1 with respect to x. This gives us:
$$\frac{du}{dx} = 2x$$
Substituting this into the previous equation, we get:
$$\frac{dy}{dx} = 3u^2 \cdot 2x = 3(x^2 + 1)^2 \cdot 2x$$
Therefore, the derivative of y with respect to x is:
$$\frac{dy}{dx} = 6x(x^2 + 1)^2$$
This is the final result, which shows how the derivative of y with respect to x can be found using the chain rule.
Implicit differentiation
What is Implicit differentiation?
Implicit differentiation is a technique that allows us to differentiate equations that cannot be expressed in the form $y = f(x)$. It is based on the fact that the derivative of a function is equal to the derivative of the dependent variable with respect to the independent variable, multiplied by the derivative of the independent variable with respect to itself.
Here is an example of implicit differentiation using KaTeX syntax:
Let's say we want to differentiate the equation $x^2 + y^2 = 9$. This equation cannot be written in the form $y = f(x)$, so we cannot use the standard rules of calculus to differentiate it. Instead, we can use implicit differentiation to find the derivative of $y$ with respect to $x$.
We start by differentiating both sides of the equation with respect to $x$:
$$\frac{d}{dx} (x^2 + y^2) = \frac{d}{dx} 9$$
Using the chain rule, we can write the derivative of the left-hand side as:
$$2x + 2y \cdot \frac{dy}{dx} = 0$$
We can then solve for $\frac{dy}{dx}$:
$$\frac{dy}{dx} = -\frac{2x}{2y} = -\frac{x}{y}$$
So, the derivative of $y$ with respect to $x$ is $-\frac{x}{y}$.
This is an example of how implicit differentiation can be used to find the derivative of a function that is not explicitly defined in terms of $x$ and $y$. It is a useful technique when we are given an equation that cannot be written in the form $y = f(x)$ and we want to find the derivative of one of the variables with respect to the other.
Derivatives
What is Derivatives?
In calculus, a derivative is a mathematical operation that describes how a function changes as its input variables (such as x in the equation f(x)) are varied. The derivative of a function at a given point is a measure of the slope of the function at that point, and it can be calculated using the following formula:
$$ \frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} $$
This formula shows that the derivative of a function is the slope of the tangent line to the function at a given point. In other words, it represents the rate at which the output of the function (y) changes as the input (x) changes.
Example of Derivatives
$$\frac{d}{dx} (\log_5 (x))^{x/2}$$
Resources
Logarithmic Differenation
What is Logarithmic Differenation?
Logarithmic differentiation is a technique that allows us to differentiate functions that cannot be directly differentiated using the standard rules of calculus. It is based on the property that the derivative of the logarithm of a function is equal to the derivative of the function divided by the function itself.
Let's say we want to differentiate the function $f(x) = x^3$. Using the power rule for derivatives, we can write the derivative as:
$$\frac{d}{dx} x^3 = 3x^2$$
Now, let's say we want to differentiate the function $g(x) = \ln(x^3)$. We can use logarithmic differentiation to find the derivative of $g(x)$ as follows:
$$\frac{d}{dx} \ln(x^3) = \frac{1}{x^3} \cdot \frac{d}{dx} x^3$$
Using the power rule, we can simplify this expression to:
$$\frac{d}{dx} \ln(x^3) = \frac{3x^2}{x^3} = \frac{3}{x}$$
So, the derivative of $\ln(x^3)$ is $\frac{3}{x}$. Note that this is the same as the derivative of $f(x)$ divided by $f(x)$ itself:
$$\frac{\frac{d}{dx} x^3}{x^3} = \frac{3x^2}{x^3} = \frac{3}{x}$$
This is the basic idea behind logarithmic differentiation. It allows us to differentiate functions that cannot be directly differentiated using the standard rules of calculus by expressing them in terms of logarithms and applying the property of the derivative of the logarithm.
Applications of Derivatives
Resources
Increasing and Decreasing Function
Increasing Function
Decreasing Function
Minimum and Maximum Values
Minimum Values
Maximum Values
Integrals
What is Integrals?
Integral calculus helps in finding the anti-derivatives of a function. These anti-derivatives are also called the integrals of the function. The process of finding the anti-derivative of a function is called integration. The inverse process of finding derivatives is finding the integrals. The integral of a function represents a family of curves. Finding both derivatives and integrals form the fundamental calculus.
Given a function, \(f\left( x \right)\), an anti-derivative of \(f\left( x \right)\) is any function \(F\left( x \right)\) such that $$F'\left( x \right) = f\left( x \right)$$ If \(F\left( x \right)\) is any anti-derivative of \(f\left( x \right)\) then the most general anti-derivative of \(f\left( x \right)\) is called an indefinite integral and denoted, $$\int{{f\left( x \right),dx}} = F\left( x \right) + c\hspace{0.25in},c{\mbox{ is an arbitrary constant}}$$ In this definition the \(\int{{}}\) is called the integral symbol, \(f\left( x \right)\) is called the integrand, \(x\) is called the integration variable and the "\(c\)" is called the constant of integration.
Example of Integral
$$ \int \sin^3(x) e^{-5x} dx $$
Resources
Calculus 2
Integration Techniques
Resources
Integrals
What is Integrals?
Integral calculus helps in finding the anti-derivatives of a function. These anti-derivatives are also called the integrals of the function. The process of finding the anti-derivative of a function is called integration. The inverse process of finding derivatives is finding the integrals. The integral of a function represents a family of curves. Finding both derivatives and integrals form the fundamental calculus.
Given a function, \(f\left( x \right)\), an anti-derivative of \(f\left( x \right)\) is any function \(F\left( x \right)\) such that $$F'\left( x \right) = f\left( x \right)$$ If \(F\left( x \right)\) is any anti-derivative of \(f\left( x \right)\) then the most general anti-derivative of \(f\left( x \right)\) is called an indefinite integral and denoted, $$\int{{f\left( x \right),dx}} = F\left( x \right) + c\hspace{0.25in},c{\mbox{ is an arbitrary constant}}$$ In this definition the \(\int{{}}\) is called the integral symbol, \(f\left( x \right)\) is called the integrand, \(x\) is called the integration variable and the "\(c\)" is called the constant of integration.
Example of Integral
$$ \int \sin^3(x) e^{-5x} dx $$
Resources
Calculus 3
Derivatives
What is Derivatives?
The derivative is the instantaneous rate of change of a function with respect to one of its variables. This is equivalent to finding the slope of the tangent line to the function at a point. Let's use the view of derivatives as tangents to motivate a geometric definition of the derivative.
The derivative of \(f\left( x \right)\) with respect to x is the function \(f'\left( x \right)\) as, $$\begin{equation}\mathop {\lim }\limits_{h \to 0} \frac{{f\left( {a + h} \right) - f\left( a \right)}}{h} \end{equation}$$
Example of Derivatives
$$\frac{d}{dx} (\log_5 (x))^{x/2}$$
Resources
Integrals
What is Integrals?
Integral calculus helps in finding the anti-derivatives of a function. These anti-derivatives are also called the integrals of the function. The process of finding the anti-derivative of a function is called integration. The inverse process of finding derivatives is finding the integrals. The integral of a function represents a family of curves. Finding both derivatives and integrals form the fundamental calculus.
Given a function, \(f\left( x \right)\), an anti-derivative of \(f\left( x \right)\) is any function \(F\left( x \right)\) such that $$F'\left( x \right) = f\left( x \right)$$ If \(F\left( x \right)\) is any anti-derivative of \(f\left( x \right)\) then the most general anti-derivative of \(f\left( x \right)\) is called an indefinite integral and denoted, $$\int{{f\left( x \right),dx}} = F\left( x \right) + c\hspace{0.25in},c{\mbox{ is an arbitrary constant}}$$ In this definition the \(\int{{}}\) is called the integral symbol, \(f\left( x \right)\) is called the integrand, \(x\) is called the integration variable and the "\(c\)" is called the constant of integration.
Example of Integral
$$ \int \sin^3(x) e^{-5x} dx $$
Resources
Integrals
What is Integrals?
Integral calculus helps in finding the anti-derivatives of a function. These anti-derivatives are also called the integrals of the function. The process of finding the anti-derivative of a function is called integration. The inverse process of finding derivatives is finding the integrals. The integral of a function represents a family of curves. Finding both derivatives and integrals form the fundamental calculus.
Given a function, \(f\left( x \right)\), an anti-derivative of \(f\left( x \right)\) is any function \(F\left( x \right)\) such that $$F'\left( x \right) = f\left( x \right)$$ If \(F\left( x \right)\) is any anti-derivative of \(f\left( x \right)\) then the most general anti-derivative of \(f\left( x \right)\) is called an indefinite integral and denoted, $$\int{{f\left( x \right),dx}} = F\left( x \right) + c\hspace{0.25in},c{\mbox{ is an arbitrary constant}}$$ In this definition the \(\int{{}}\) is called the integral symbol, \(f\left( x \right)\) is called the integrand, \(x\) is called the integration variable and the "\(c\)" is called the constant of integration.
Example of Integral
$$ \int \sin^3(x) e^{-5x} dx $$
Resources
Integrals
What is Integrals?
Integral calculus helps in finding the anti-derivatives of a function. These anti-derivatives are also called the integrals of the function. The process of finding the anti-derivative of a function is called integration. The inverse process of finding derivatives is finding the integrals. The integral of a function represents a family of curves. Finding both derivatives and integrals form the fundamental calculus.
Given a function, \(f\left( x \right)\), an anti-derivative of \(f\left( x \right)\) is any function \(F\left( x \right)\) such that $$F'\left( x \right) = f\left( x \right)$$ If \(F\left( x \right)\) is any anti-derivative of \(f\left( x \right)\) then the most general anti-derivative of \(f\left( x \right)\) is called an indefinite integral and denoted, $$\int{{f\left( x \right),dx}} = F\left( x \right) + c\hspace{0.25in},c{\mbox{ is an arbitrary constant}}$$ In this definition the \(\int{{}}\) is called the integral symbol, \(f\left( x \right)\) is called the integrand, \(x\) is called the integration variable and the "\(c\)" is called the constant of integration.
Example of Integral
$$ \int \sin^3(x) e^{-5x} dx $$