The indefinite integral

Differentiation has a natural question attached to it: given a rate of change, what was the original function? That reversal is antidifferentiation, and its answer is never a single function. Because the derivative of any constant is zero, x² and x² + 7 and x² − 100 all have derivative 2x. Recovering "the" antiderivative of 2x therefore recovers a whole family, written x² + C.

The +C is not decoration. It is the degree of freedom that a rate of change cannot determine, and pinning it down needs one extra piece of information — an initial position, a starting temperature, a boundary condition. Drop it and you have silently assumed one particular member of the family, which is how a correct integration turns into a wrong physical answer.

Integration is also much harder than differentiation, in a specific way. There are product and quotient rules for derivatives, but no corresponding formulas for integrals — ∫fg is not ∫f·∫g, and problems like ∫x·cos x dx need integration by parts or a substitution, chosen by judgement rather than applied mechanically. Worse, some perfectly ordinary functions, e^(−x²) among them, have no antiderivative expressible in elementary terms at all. That is not a gap in anyone's technique; it is a proven limitation, and it is why numerical integration exists.