Business Mathematics and Statistics · Ch 7 — Integration

Integration as the Inverse of Differentiation

The CHSE Odisha (Council of Higher Secondary Education) Class 11 Commerce syllabus places Integration directly after Differentiation in the Business Mathematics and Statistics elective — and for good reason: integration is differentiation run in reverse. 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 opposite question: which function, when differentiated, gives this one back?

Antiderivative (Primitive)

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 — $x^{3}+7$ , $x^{3}-10$ and $x^{3}+50$ all differentiate to the same $3x^{2}$ , because the derivative of any constant is zero. An antiderivative is therefore never unique.

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 $C$ , the constant of integration, is an arbitrary real number. The symbol $\int \cdots dx$ is read "the integral of ... with respect to $x$ "; $f(x)$ is called 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. $\int 3x^{2}\,dx = x^{3}+C$ stands for the WHOLE family of curves $x^{3}+C$ , one for every value of $C$ — not a single curve. Dropping $C$ is treated as a genuine error in this chapter's worked examples, never a rounding-off shortcut.

Checking an integral by differentiating back

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

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

This dual-solve habit — solve the integral, then differentiate the result to verify — catches an arithmetic slip far more reliably than re-reading the same working a second time, and it is used throughout this chapter's worked examples without exception, even when an integral looks routine.

The CHSE Odisha Std-11 Business Mathematics and Statistics syllabus draws on the same mathematical principles of integral calculus taught, under slightly different notation and emphasis, across other Indian boards' commerce-mathematics curricula — only the applied business context in the closing section of this chapter is specific to this syllabus.

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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