Business Mathematics and Statistics · Ch 2 — Integral Calculus – I (Indefinite/Definite Integrals)

Indefinite Integration — Standard Integrals

This Tamil Nadu HSC Class 12 Business Mathematics and Statistics chapter introduces integration — the reverse process of differentiation — building the toolkit that the next chapter uses for genuine business/economics applications like total cost, total revenue, and area under a demand curve.

Integration as the reverse of differentiation

If $\frac{d}{dx}F(x)=f(x)$ , then $F(x)$ is called an antiderivative (or indefinite integral) of $f(x)$ , written

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

where $C$ is an arbitrary constant of integration — because the derivative of any constant is zero, $F(x)+C$ is equally valid for EVERY value of $C$ , so a single antiderivative is never the full answer without it.

Standard integrals

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

\int e^x\,dx=e^x+C\qquad \int a^x\,dx=\frac{a^x}{\ln a}+C\ (a>0,a\neq1)

\int \cos x\,dx=\sin x+C\qquad \int \sin x\,dx=-\cos x+C

Note

Always verify by differentiating back

The fastest way to check any integration is to differentiate the answer and confirm it reproduces the original integrand — this works because integration and differentiation are exact inverses of one another, and it catches almost every arithmetic slip.

These standard forms, applied term-by-term to a sum (since $\int[f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx$ ), already handle every polynomial integral — the substitution and by-parts techniques in the next section extend this toolkit to products and compositions of functions.

Definition 1Indefinite Integral

$\int f(x)\,dx=F(x)+C$ , where $F'(x)=f(x)$ and $C$ is an arbitrary constant of integration.

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