Introduction

Calculus is the study of change.

In both of these branches (Differential and Integral), the concepts learned in algebra

and geometry are extended using the idea of limits. Limits allow us to study what happens

when points on a graph get closer and closer together until their distance is infinitesimally

small (almost zero). Once the idea of limits is applied to our Calculus problem, the

techniques used in algebra and geometry can be implemented.

Differential calculus and integral calculus are connected by the fundamental

theorem of calculus, which states that differentiation is the reverse process to integration.

Leibniz’s Creation of the Calculus

In the years from 1672 to 1676, spent in Paris, Leibniz’s slowly flowering

mathematical genius matured. During this time, he developed the principal features and

notation of his version of the Calculus. Various methods had been invented for determining

the tangent lines to certain classes of curves, but as yet nobody had made known similar

procedures for solving the inverse problem, that is deriving the equation of the curve itself

from the properties of the tangents. Leibniz stated the inverse tangent problem thus: “To

find the locus of the function, provided the locus which determines the subtangent is

known.” By the middle of 1673, he had settled down to an exploration of this problem, fully

recognizing that “almost the whole of the theory of the inverse method of tangents is

reducible to quadratures [integrations].”

Because Leibniz was still struggling with the notation for his calculus, it is not

surprising that these early calculations were clumsy. Either he expressed his results in

rhetorical form or else used abbreviations, such as “omn.” For the Latin omnia (“all”) to

mean “sum.” The letter l was used to symbolize what we should write as dy, the

“difference” of two neighboring ordinates .

Differential Calculus

In mathematics, differential calculus is a subfield of calculus concerned with the

study of the rates at which quantities change. It is one of the two traditional divisions of

calculus, the other being integral calculus.

The primary objects of study in differential calculus are the derivative of a function,

related notions such as the differential, and their applications. The derivative of a function

at a chosen input value describes the rate of change of the function near that input value.

The process of finding a derivative is called differentiation. Geometrically, the derivative at a

point is the slope of the tangent line to the graph of the function at that point, provided that

the derivative exists and is defined at that point. For a real-valued function of a single real

variable, the derivative of a function at a point generally determines the best linear

approximation to the function at that point.

Integral Calculus

The integral is an important concept in mathematics. Integration is one of the two

main operations in calculus, with its inverse, differentiation, being the other. Given a

function f of a real variable x and an interval [a, b] of the real line, the definite integral

is defined informally as the signed area of the region in the xy-plane that is bounded

by the graph of f, the x-axis and the vertical lines x = a and x = b. The area above the x-axis

adds to the total and that below the x-axis subtracts from the total.

The term integral may also refer to the related notion of the antiderivative, a

function F whose derivative is the given function f. In this case, it is called an indefinite

integral and is written:

What is the difference between these two?

Differential Calculus cuts something into small pieces to find how it changes.

Integral Calculus joins (integrates) the small pieces together to find how much there

is.

Antidifferentiation

Antidifferentiation is the process of finding the set of all the antiderivatives of a

given function. The symbol ∫ denotes the operation of antidifferentiation, and we write

∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥) + 𝐶

where 𝐹 ′(𝑥) = 𝑓(𝑥) and 𝑑(𝐹(𝑥)) = 𝐹(𝑥) + 𝐶. The expression 𝐹(𝑥) + 𝐶 is the general

antiderivative of f.

Leibniz introduced the convention of writing the differential of a function after the

antidifferentiation symbol. The advantage of using the differential in this manner will be

apparent when we compute antiderivatives by changing the variable. From the above

equations, we can write:

∫ 𝑑(𝐹(𝑥)) = 𝐹(𝑥) + 𝐶

This equation states that when we antidiffentiate the differential of a function, we obtain

that function plus an arbitrary constant. So we can think of the ∫ symbol for

antidifferentiation as meaning that operation which is the inverse of the operation denoted

by d for computing differential.

If { 𝐹(𝑥) + 𝐶} is the set of all functions whose differentials are𝑓(𝑥)𝑑𝑥, it is also the

set of all functions whose derivatives are f(x). Antidifferentiation, therefore, is considered as

the operation of finding the set of all functions having a given derivative.

Notation for Antiderivatives

The antidifferentiation process is also called integration and is denoted by the

symbol ∫ called an integral sign. The symbol ∫ 𝑓(𝑥)𝑑𝑥 is called the indefinite integral of f(x),

and it denotes the family of antiderivatives of f(x). That is, if 𝐹 ′(𝑥) = 𝑓(𝑥) for all x, then

∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥) + 𝐶 where f(x) is called the integrand and C the constant of integration.

The differential dx in the indefinite integral identifies the variable of integration. That

is, the symbol ∫ 𝑓(𝑥)𝑑𝑥 denotes the “antiderivative of f with respect to x” just as the symbol

dy/dx denotes the “derivative of y with respect to x.”

Integration Tables

Basic Integration Formulas

Rules Antiderivatives

1. Constant Rule ∫ 𝑑𝑥 = 𝑥 + 𝐶

2. Simple Power Rule ∫ 𝑥 𝑛 𝑑𝑥 =

𝑥 𝑛+1

𝑛 + 1+ 𝐶, 𝑛 ≠ −1

3. General Power Rule ∫ 𝑢𝑛

𝑑𝑢

𝑑𝑥𝑑𝑥 = ∫ 𝑢𝑛 𝑑𝑢 =

𝑢𝑛+1

𝑛 + 1+ 𝐶, 𝑛 ≠ −1

4. Simple Log Rule ∫

1

𝑥𝑑𝑥 = ln|𝑥| + 𝐶

5. General Log Rule ∫

1

𝑢 𝑑𝑢

𝑑𝑥𝑑𝑥 = ∫

1

𝑢 𝑑𝑢 = ln|𝑢| + 𝐶

6. Simple Exponential Rule ∫ 𝑒𝑥 𝑑𝑥 = 𝑒𝑥 + 𝐶

7. General Exponential Rule ∫ 𝑒𝑢

𝑑𝑢

𝑑𝑥𝑑𝑥 = ∫ 𝑒𝑢 𝑑𝑢 = 𝑒𝑢 + 𝐶

Trigonometric Functions Integration Formulas

Functions Antiderivatives

1. ∫ 𝑐𝑜𝑠𝑥 𝑑𝑥 𝑠𝑖𝑛𝑥 + 𝐶

2. ∫ 𝑠𝑖𝑛𝑥 𝑑𝑥 −𝑐𝑜𝑠𝑥 + 𝐶

3. ∫ 𝑠𝑒𝑐2 𝑥 𝑑𝑥 𝑡𝑎𝑛𝑥 + 𝐶

4. ∫ 𝑐𝑠𝑐2 𝑥 𝑑𝑥 −𝑐𝑜𝑡𝑥 + 𝐶

5. ∫ 𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥 𝑑𝑥 𝑠𝑒𝑐𝑥 + 𝐶

6. ∫ 𝑐𝑠𝑐𝑥𝑐𝑜𝑡𝑥 𝑑𝑥 −𝑐𝑠𝑐𝑥 + 𝐶

Sample Problems Solving the Antiderivatives of Basic Equation

Sample Problems Solving the Antiderivatives of Trigonometric Functions

Differential Equations

A differential equation is an equation involving a function and one or more of its

derivatives. For instance,

3𝑑𝑦

𝑑𝑥− 2𝑥𝑦 = 0

is a differential equation in which y = f(x) is a differentiable function of x.

A function y = f(x) is called a solution of a given differential equation if the equation is

satisfied when y and its derivatives are replaced by f(x) and its derivatives. For example,

𝑦 = 𝑒−2𝑥

is a solution to the differential equation

𝑦′ + 2𝑦 = 0

because 𝑦′ = −2𝑒−2𝑥 , and by substitution we have

−2𝑒−2𝑥 + 2𝑒−2𝑥 = 0

Furthermore, we can readily see that

𝑦 = 2𝑒−2𝑥 , 𝑦 = 3𝑒−2𝑥 , 𝑦 = 1

2𝑒−2𝑥

are also solutions to the same differential equation. In fact, the functions

𝑦 = 𝐶𝑒−2𝑥

where C is any real number, are all solutions. We call this family of solutions the general

solution of the differential equation 𝑦′ + 2𝑦 = 0.

A particular solution of a differential equation is any solution that is obtained by

assigning specific values to the constants in the general solution.

Geometrically, the general solution of a given differential equation represents a

family of curves, called solution curves one for each value assigned to the arbitrary

constants.

Example of Differential Equations (First and Second Order)

1. Show that 𝑑2 𝑦

𝑑𝑥2 = 2𝑑𝑦

𝑑𝑥 a solution of y = c1 + c2e2x

Solution:

Since y = c1 + c2e2x, then:

𝑑𝑦

𝑑𝑥= 2c2e2x

and

2𝑑𝑦

𝑑𝑥= 4c2e2x

Therefore, 𝑑2 𝑦

𝑑𝑥2 = 2𝑑𝑦

𝑑𝑥.

References

Books

Burton, David M. The History of Mathematics

Larson and Hostetler. Brief Calculus with Applications. “Integration.” Fourth Printing.

P 316-518

Leithold, Louis. TC 7 (The Calculus 7). P 315-32

Websites

http://en.wikipedia.org/wiki/Calculus

http://en.wikipedia.org/wiki/Integral

http://en.wikipedia.org/wiki/Differential_calculus

http://www.mathsisfun.com/calculus/integration-introduction.html

http://tutorial.math.lamar.edu/Classes/CalcII/IntegralsWithTrig.aspx