Integration and Antiderivatives

Undoing the Derivative

Integration is the reverse of differentiation. An antiderivative of $f$ is a function $F$ where $F'(x) = f(x)$ , recovering original totals from rates of change.

Antiderivatives are not unique. Because the derivative of a constant is zero, adding $C$ yields the indefinite integral:

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

Case	Rule	Pitfall
General	$\int x^n\, dx = \frac{x^{n+1}}{n+1} + C$ ( $n \neq -1$ )	Raising the power, not lowering
Special	$\int \frac{1}{x}\, dx = \ln\|x\| + C$	Fails when $n = -1$

Common pitfall: Forgetting the constant of integration $+C$ . Infinitely many functions share the same derivative, differing only by a vertical shift.

Integration's power rule is differentiation's, run backward ---

Rules and Techniques

Integration rules directly reverse standard differentiation rules. Constants factor out, and the integral of a sum equals the sum of the integrals.

Function $f(x)$	Indefinite Integral $\int f(x)\, dx$
$\cos x$	$\sin x + C$
$e^x$	$e^x + C$
$\frac{1}{x}$	$\ln\|x\| + C$

More complex integrals require advanced tools:

Substitution: Reverses the chain rule.
Integration by parts: Reverses the product rule.

Common pitfall: Applying the derivative power rule (bringing the exponent down) instead of the integration power rule (raising the exponent and dividing).

Undoing the Derivative

Rules and Techniques

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Calculus