Antiderivatives and Indefinite Integrals

Undoing Derivatives

Integration reverses differentiation. An antiderivative of $f$ is a function $F$ where $F'(x) = f(x)$ . Where derivatives ask for rates of change, antiderivatives recover the original function from its rate.

Antiderivatives are not unique. Since any constant has a zero derivative, if $F$ is an antiderivative, then $F + C$ is also one. The indefinite integral represents this entire family of parallel curves:

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

Rule	Formula	Notes
Power Rule	$\int x^n\, dx = \frac{x^{n+1}}{n+1} + C$	For $n \neq -1$
Reciprocal	$\int \frac{1}{x}\, dx = \ln\|x\| + C$	Handles $n = -1$
Exponential	$\int e^x\, dx = e^x + C$	Unchanged
Cosine	$\int \cos x\, dx = \sin x + C$	Reverses sine derivative

A family of parabolas, all parallel where it counts

Rules and Pitfalls

Linearity rules carry over directly from differentiation: constants factor out, and the integral of a sum is the sum of the individual integrals.

More advanced techniques like substitution (reversing the chain rule) and integration by parts (reversing the product rule) handle complex cases later.

Common Mistake	Why It Happens	Correct Approach
Omitting $+C$	Treating the integral as a single static function	Always append $+C$ to represent the infinite vertical family
Power Rule Slip	Differentiating instead of integrating	Raise the exponent and divide by the new power

Undoing Derivatives

Rules and Pitfalls

Practise this lesson

Integral Calculus