Indefinite Integrals and Basic Integration Techniques

This class introduces the basic techniques to compute the most elementary indefinite integrals, as well as the properties of the integration operator. This includes polynomial, exponential, hyperbolic, and basic trigonometric integrals.

Learning Objectives:
By the end of this class, the student will be able to

1. Understand the process of indefinite integration as the inverse process of differentiation.
2. Compute the integral of polynomials and expressions involving exponential, hyperbolic, and trigonometric functions.
3. Use the properties of integrals to perform algebraic manipulations that facilitate their computation.

TABLE OF CONTENTS
THE RELEVANCE OF INDEFINITE INTEGRALS
ANTIDERIVATIVES, INDEFINITE INTEGRALS, AND FUNCTION PRIMITIVES
BASIC INTEGRATION TECHNIQUES

The Relevance of Indefinite Integrals

Indefinite integrals are a fundamental tool in calculus and have a wide range of applications in physical and mathematical sciences. They allow us to compute the primitive function of a given function, which in turn is used to calculate areas under curves, volumes of solids, probabilities, and many other applications in physics, engineering, statistics, and economics. Furthermore, indefinite integrals are essential for solving differential equations, making them indispensable in many fields of science and technology.

Antiderivatives, Indefinite Integrals, and Function Primitives

If a function F(x) has derivative f(x) on a given interval I, then F(x) is said to be a primitive of f(x) on that interval.

It is important to note that if F(x) is a primitive of f(x), then so is F(x) + C, where C is any real constant. This is represented by writing:

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

The constant C is known as the constant of integration, and its presence indicates that the primitive of a function is not a single function but a family of functions: the set of all functions whose derivative is f(x) on the interval I.

The words antiderivative, primitive, and indefinite integral are three ways to express the same idea, so we use them interchangeably. In summary, the indefinite integral is the inverse process of differentiation, and from this idea its most fundamental properties are derived.

Basic Properties of Indefinite Integrals

In order to compute indefinite integrals, we must first know some basic properties, which are directly inherited from the properties of derivatives.

1. \displaystyle \int \dfrac{df(x)}{dx} dx = f(x) + C
   Because the indefinite integral is the inverse process of differentiation.




2. \displaystyle \int \lambda f(x) dx = \lambda \int f(x) dx
   Where \lambda is any real constant. This holds because
   
   \begin{array} {} \displaystyle \int \lambda \dfrac{d\phi(x)}{dx}dx &= \displaystyle \int \dfrac{d}{dx}\lambda \phi(x) dx \\ \\ &= \lambda \phi(x) + C_1 \\ \\ &= \lambda(\phi(x) + C_2) \\ \\ &= \lambda \displaystyle \int \frac{d\phi(x)}{dx}dx \end{array}
   
   
   And then, using f(x) = \dfrac{d\phi(x)}{dx} we get
   
   \displaystyle \int \lambda f(x) dx = \lambda \int f(x)dx




3. \displaystyle \int f(x) + g(x) dx = \int f(x) dx + \int g(x) dx
   
   This can be demonstrated in a similar way. Consider two functions \phi(x) and \psi(x) such that
   
   
   f(x) = \dfrac{d\phi(x)}{dx} and g(x) = \dfrac{d\psi(x)}{dx}
   
   
   Then we have
   
   
   \begin{array} {} \displaystyle \int f(x) + g(x) dx &= \displaystyle \int \dfrac{d\phi(x)}{dx} + \dfrac{d\psi(x)}{dx} dx \\ \\ &= \displaystyle \int \dfrac{d}{dx} (\phi(x) + \psi(x)) dx \\ \\ &= \phi(x) + \psi(x) + C \\ \\ &= (\phi(x) + C_1) + (\psi(x) + C_2) \\ \\ &= \displaystyle \int \dfrac{d\phi(x)}{dx} dx + \int \dfrac{d\psi(x)}{dx}dx \\ \\ &= \displaystyle \int f(x) dx + \int g(x) dx \end{array}

Basic Integration Techniques

There are basic integration techniques that allow us to compute some indefinite integrals using results obtained from differentiation. Through these techniques, we can obtain the following useful results for integration:

Integrals of Polynomial Functions

1. \displaystyle \int 1 dx = x + C
   
   Because \dfrac{d}{dx} (x + C)= 1
2. \displaystyle \int x^q dx = \dfrac{x^{q+1}}{q+1} + C, provided that q\neq -1
   
   Because \dfrac{d}{dx} \left(\dfrac{x^{q+1}}{q+1} + C\right) = x^q.

With these results and the basic properties, we can compute the integral of any polynomial without difficulty.

Exponential and Logarithmic Integrals

From the known results of the derivatives of exponential and logarithmic functions, we obtain the following basic results:

1. \displaystyle \int e^{x}dx = e^{x} + C
   Because \dfrac{d}{dx}\left(e^x + C\right) = e^x
2. \displaystyle \int \dfrac{1}{x} dx = ln|x| + C
   
   Because \dfrac{d}{dx}\left(ln|x| + C \right) = \dfrac{1}{|x|} sig(x) = \dfrac{1}{x}
   
   Where sig(x) is the sign function defined as follows:
   
   
   sig(x) = \left\{\begin{array}{} +1 &,&0\lt x \\ -1 &,& x\lt 0 \end{array}\right.

The result of the integral of 1/x allows us to expand our ability to integrate functions, as we can begin integrating functions that are quotients of polynomials.

Integrals of Basic Hyperbolic Functions

The basic hyperbolic functions are

\begin{array} {} sinh(x) &=& \dfrac{e^x - e^{-x}}{2} \\ \\ cosh(x) &=& \dfrac{e^x + e^{-x}}{2} \end{array}

Since we have already seen how the exponential function integral works, we will have no problem with the integrals of hyperbolic sine and cosine.

For hyperbolic sine the computation is practically direct:

\begin{array} {} \displaystyle \int sinh(x) dx &=& \displaystyle \int \dfrac{e^x - e^{-x}}{2}dx \\ \\ &=& \dfrac{1}{2} \left( \displaystyle \int e^x dx - \int e^{-x} dx \right) \\ \\ &=& \dfrac{1}{2} \left(e^x + e^{-x} \right) + C = cosh(x) + C \end{array}

And for hyperbolic cosine, the calculations are practically analogous:

\begin{array} {} \displaystyle \int cosh(x) dx &=& \displaystyle \int \dfrac{e^x + e^{-x}}{2}dx \\ \\ &=& \dfrac{1}{2} \left( \displaystyle \int e^x dx + \int e^{-x} dx \right) \\ \\ &=& \dfrac{1}{2} \left(e^x - e^{-x} \right) + C = sinh(x) + C \end{array}

Besides these, there are many other hyperbolic functions that can be integrated:

\begin{array} {} tanh(x) &=& \dfrac{sinh(x)}{cosh(x)} \\ sech(x) &=& \dfrac{1}{cosh(x)} \\ {}csch(x) &=& \dfrac{1}{sinh(x)} \\ ctgh(x) &=& \dfrac{1}{tanh(x)} \end{array}

However, integrating them requires other techniques that we will explore in future classes.

Integrals of Basic Trigonometric Functions

The basic trigonometric functions are sin(x) and cos(x). Computing their integrals is practically direct based on what we already know from their derivatives.

\begin{array} {} \displaystyle \int sin(x) dx = -cos(x) + C \\ \\ {} \displaystyle \int cos(x) dx = sen(x) + C \end{array}

This occurs because

\begin{array} {} \dfrac{d}{dx}\left( sin(x) + C \right) &=& cos(x) \\ \\ {} \dfrac{d}{dx}\left( cos(x) + C \right) &=& -sin(x) \\ \\ \end{array}

Conclusion

In this class, we have explored indefinite integrals from their theoretical foundations to their most basic practical applications. We have learned to recognize them as the inverse process of differentiation, to identify their basic properties, and to apply direct techniques to integrate simple polynomial, exponential, logarithmic, hyperbolic, and trigonometric functions. This knowledge forms the essential foundation for tackling more complex integration problems in the future and will be fundamental for studying advanced applications in physics, engineering, and other sciences. With this foundational knowledge, we will be able to introduce more sophisticated techniques in future classes.