SMA 200 • UNIT 2 • Mathematics
Integration Of Functions Using The Extension Of The Basic
Topics covered: Objective of the Lesson, Definition of Integrand, The Extension of the Basic Formulae, Proof of the Extension Formulae
Syllabus Study Notes (Core Overview)
Free Overview Preview1. Core Concept Overview & Intuition
Integration is defined as the reverse or converse process of differentiation. In differential calculus one begins with a function and obtains its derivative; in integral calculus one begins with a derivative and seeks the function from which it arose. For instance, since the derivative of $\sin x$ is $\cos x$, one says that the integral of $\cos x$ is $\sin x$, written $\int \cos x\,dx=\sin x$.
The basic formulae of the preceding unit handle integrands that are elementary power, exponential, and trigonometric functions in their simplest form. The present unit extends those formulae to integrands in which the variable appears in a linear combination, that is, expressions of the form $ax+b$ where $a$ and $b$ are constants with $a\neq 0$. The central intuition is that a linear change of variable inside a function rescales the antiderivative by the reciprocal of the coefficient of the variable. Thus every basic formula acquires a companion version in which $x$ is replaced by $ax+b$ and the result is divided by $a$. This single structural idea converts ten basic formulae into a much wider family of immediately integrable expressions, and it is the foundation on which later techniques of substitution are built.
Because any constant added to an antiderivative disappears under differentiation, every indefinite integral carries an arbitrary constant $c$, called the constant of integration. The symbol $\int$ was introduced by Leibnitz, who took it from the first letter $S$ of the word summation; it is pronounced "the integration of". The integral sign cannot be divorced from the differential $dx$ when the integration is with respect to $x$.
2. Mathematical Formalization & Core Definitions
> Definition (Integration). Integration is the reverse or converse process of differentiation. If $\frac{d}{dx}F(x)=f(x)$, then $\int f(x)\,dx=F(x)+c$, where $c$ is an arbitrary constant.
> Definition (Constant of Integration). Since $\frac{d}{dx}(\sin x)=\frac{d}{dx}(\sin x+c)=\cos x$ for any constant $c$, a constant is always appended after the process of integration. The symbol $c$ is called the constant of integration.
> Definition (Integrand). If $f(x)$ is any function of $x$, then $\int f(x)\,dx$ stands for the integral of $f(x)$ with respect to $x$. The function $f(x)$ appearing in $\int f(x)\,dx$ is called the integrand of the integral.
The integrand is therefore the function being integrated, that is, the expression standing between the integral sign and the differential. For example, in $\int x^2\,dx$ the integrand is $x^2$; in $\int \cos x\,dx$ the integrand is $\cos x$; in $\int 2^x\,dx$ the integrand is $2^x$. The differential $dx$ indicates the variable of integration and is not part of the integrand.
The basic formulae established in the preceding unit are the following.
| No. | Basic Formula | Class |
|---|---|---|
| 1 | $\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+c,\quad n\neq -1$ | Algebraic |
| 2 | $\int \dfrac{1}{x}\,dx=\int x^{-1}\,dx=\ln x+c$ | Algebraic |
| 3 | $\int a^x\,dx=\dfrac{a^x}{\ln a}+c,\quad a>0$ | Exponential |
| 4 | $\int e^x\,dx=e^x+c$ | Exponential |
| 5 | $\int \cos x\,dx=\sin x+c$ | Trigonometric |
| 6 | $\int \sin x\,dx=-\cos x+c$ | Trigonometric |
| 7 | $\int \sec^2 x\,dx=\tan x+c$ | Trigonometric |
| 8 | $\int \sec x\tan x\,dx=\sec x+c$ | Trigonometric |
| 9 | $\int \operatorname{cosec} x\cot x\,dx=-\operatorname{cosec} x+c$ | Trigonometric |
Each trigonometric formula is justified by differentiating the right-hand side, for example $\frac{d}{dx}(\tan x+c)=\sec^2 x$ and $\frac{d}{dx}(\sec x+c)=\sec x\tan x$.
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