Mathematics and Statistics · Ch 5 — Integration

Integration as the Anti-derivative

In the earlier chapters of Maharashtra Board Std XII Commerce Mathematics and Statistics, differentiation took a function and produced its rate of change. Integration reverses that operation: it starts from a function and asks which function, when differentiated, gives this one back? This chapter builds the whole idea of integration as an anti-derivative from that single reverse question. (Throughout this chapter, $\log$ denotes the natural logarithm, to base $e$ .)

Anti-derivative (primitive)

A function $F(x)$ is called an anti-derivative (or primitive) of $f(x)$ if

F'(x) = f(x).

For example, since $\dfrac{d}{dx}\left(x^{4}\right) = 4x^{3}$ , the function $x^{4}$ is an anti-derivative of $4x^{3}$ . But it is not the only one: $x^{4}+7$ , $x^{4}-2$ and $x^{4}+100$ all differentiate to the same $4x^{3}$ , because the derivative of any constant is zero. So an anti-derivative is never unique — any two anti-derivatives of the same function differ only by a constant.

The indefinite integral and the constant of integration

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

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

where $F(x)$ is any one anti-derivative 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 $x$ "; the function $f(x)$ being integrated is the integrand.

Note

Why the constant $c$ can never be dropped

Differentiating removes a constant term completely (the derivative of a constant is $0$ ), so integrating cannot recover which constant was there. Writing $\displaystyle\int 4x^{3}\,dx = x^{4}+c$ represents the WHOLE family of curves $x^{4}+c$ for every value of $c$ — not one single curve. Omitting $c$ from an indefinite integral is treated as a genuine error, not a shortcut.

Verifying 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 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, and it is used as an explicit check in every worked example that follows. Maharashtra's Std XII Commerce Mathematics and Statistics syllabus develops integration from the same standard calculus principles taught in Class 12 commerce-mathematics curricula across India — the notation and standard results below are the common core of that treatment.

Definition 1Anti-derivative (Primitive)

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

Definition 2Indefinite Integral

$\displaystyle\int f(x)\,dx = F(x)+c$ — the family of ALL anti-derivatives of $f(x)$ , where $F(x)$ is one anti-derivative and $c$ is the arbitrary constant of integration.

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