Business Mathematics and Basic Statistics · Ch 14 — Integration

Integration as the Inverse of Differentiation

Business Mathematics and Basic Statistics (WBCHSE Class 12 Commerce, Semester IV) now turns the Limits and Derivatives chapter's central operation around. Differentiation started from a function and found its rate of change. Integration starts from a rate of change (or from any function) and asks the reverse question: which function, when differentiated, gives this one back?

Antiderivative — the reverse of a derivative

A function $F(x)$ is called an antiderivative (or primitive) of $f(x)$ if

F'(x) = f(x)

For example, since $\dfrac{d}{dx}(x^{3}) = 3x^{2}$ , the function $x^{3}$ is an antiderivative of $3x^{2}$ . But it is not the ONLY one — $x^{3}+5$ , $x^{3}-2$ , and $x^{3}+100$ all differentiate to the same $3x^{2}$ , because the derivative of any constant is zero. This is why an antiderivative is never unique, and why integration always carries an extra term to account for it.

The indefinite integral and the constant of integration

The collection of ALL antiderivatives of $f(x)$ is written as the indefinite integral:

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

where $F(x)$ is any one antiderivative of $f(x)$ and $C$ is an arbitrary constant, called the constant of integration. The symbol $\int \cdots dx$ is read "the integral of $\ldots$ with respect to $x$ "; $f(x)$ is the integrand.

Note

Why $C$ can never be dropped

Differentiating REMOVES a constant term completely (the derivative of any constant is $0$ ), so integrating cannot possibly recover it. $\displaystyle\int 3x^{2}\,dx = x^{3}+C$ represents the WHOLE family of curves $x^{3}+C$ for every possible value of $C$ — not one single curve. Dropping $C$ is treated as a genuine error in this chapter's worked examples and exercises, not a rounding-off convenience.

Checking an integral by differentiating back

Because integration undoes differentiation, every indefinite-integral answer in this chapter can be — and should be — checked by differentiating the answer and confirming the ORIGINAL integrand reappears:

\frac{d}{dx}\Big[F(x)+C\Big] = f(x)

This "differentiate the result to verify" habit is the single most reliable way to catch an arithmetic slip in an integration problem, and it is used throughout this chapter's worked examples exactly the way a dual check should be used — never skipped, even when the integral itself looks routine.

WBCHSE's Business Mathematics and Basic Statistics syllabus keeps this chapter's scope to the standard-formula integrals, a restricted form of integration by parts, the statement (not the proof) of the Fundamental Theorem of Integral Calculus, and simple definite-integral evaluations — the same core integration techniques taught, under different syllabus wording, in commerce-mathematics and general-mathematics curricula across India.

Definition 1Antiderivative (Primitive)

A function $F(x)$ is an antiderivative of $f(x)$ if $F'(x) = f(x)$ . An antiderivative is never unique — adding any constant to it gives another antiderivative of the same function.

Definition 2Indefinite Integral

$\displaystyle\int f(x)\,dx = F(x)+C$ — the family of ALL antiderivatives of $f(x)$ , where $F(x)$ is one antiderivative and $C$ is the arbitrary constant of integration.

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