Mathematics · Ch 10 — Integrals

Integration as the Inverse Process of Differentiation

Differentiation takes a function $f(x)$ to its derivative $f'(x)$ ; integration reverses this process -- given a function, it recovers a function whose derivative is the given one. This is

why integration is also called antidifferentiation, and an integral in this form (with no

upper/lower limits) is called an indefinite integral.

Definition (antiderivative). A function $F(x)$ is called an antiderivative (or

primitive) of $f(x)$ if

\frac{d}{dx}\big[F(x)\big] = f(x)

for every $x$ in the domain under consideration. Since $\dfrac{d}{dx}\big[F(x)+C\big]=f(x)$ for

any constant $C$ (the derivative of a constant is $0$ ), a function does not have a single

antiderivative but an entire family of them, all differing by a constant. This whole family

is written as the indefinite integral

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

where $C$ is called the constant of integration, and $f(x)$ is the integrand.

Geometric meaning. The graphs of $y=F(x)+C$ for different values of $C$ form a family of

curves, each a vertical translate of the others; at any fixed $x$ -value, every curve in the

family has exactly the same slope $f(x)$ , since they all share the same derivative. Fixing one

point the curve must pass through pins down the value of $C$ .

Standard integrals, each obtained directly by reversing a known derivative:

\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\ (n\neq-1), \quad \int\frac{dx}{x}=\ln|x|+C, \quad \int e^x\,dx=e^x+C,

\int\sin x\,dx=-\cos x+C, \quad \int\cos x\,dx=\sin x+C, \quad \int\sec^2x\,dx=\tan x+C,

\int\frac{dx}{\sqrt{1-x^2}}=\sin^{-1}x+C, \qquad \int\frac{dx}{1+x^2}=\tan^{-1}x+C.

Linearity. For constants $a,b$ ,

\int\big[af(x)+bg(x)\big]\,dx = a\int f(x)\,dx + b\int g(x)\,dx,

mirroring the linearity of differentiation -- this is what allows a polynomial or a sum of

several standard terms to be integrated term by term.

The definitive check. Because integration is exactly the reverse of differentiation, every

indefinite integral computed in this chapter can -- and should -- be verified by differentiating

the answer and confirming it reproduces the original integrand exactly. This is the single most

reliable way to catch an algebra slip or a wrong sign, and Example 1 demonstrates it directly:

differentiating the claimed antiderivative $x^3+x^2$ recovers $3x^2+2x$ , confirming

$\int(3x^2+2x)\,dx=x^3+x^2+C$ .

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