Differentiation and integration are inverse operations, but they don't feel symmetrical to work with. Given essentially any function built from standard pieces, differentiation offers a fixed, mechanical set of rules — the power rule, product rule, chain rule, and a handful of others — that, applied correctly and patiently, always produces the derivative. Integration offers no equivalent guaranteed procedure. Techniques like substitution, integration by parts, and partial fractions are genuinely useful tools, but which one to try, and whether any of them will actually work, isn't something a fixed algorithm can always determine in advance.
Differentiation is mechanical because it only ever gets simpler
Differentiation's rules work reliably because differentiating a function, piece by piece according to its structure, systematically breaks it down into simpler component pieces, and a fixed set of rules covers how each standard type of piece differentiates and how those pieces combine. Given enough patience, differentiation of a function built from familiar standard components always terminates in a definite answer, following the same predictable procedure every time — which is exactly why differentiation, unlike integration, can genuinely be automated as a fully mechanical, guaranteed-to-succeed process.
Integration doesn't have the equivalent guarantee
Integration is the reverse process, and reversing a process is often structurally harder than performing it forward — much like it's easy to multiply two known numbers together but comparatively harder to factor a large number back into its original components. Techniques like substitution (recognising a function as the result of a chain rule differentiation, and undoing that structure), and integration by parts (undoing a product rule differentiation), each work only when a given integral happens to match the specific pattern that technique is built to reverse. A skilled calculus student learns to recognise which pattern a given integral resembles and try the matching technique, but there's no fixed, guaranteed sequence of steps that's certain to find an answer for an arbitrary integral the way differentiation's rules are certain to find a derivative — and, notably, some integrals of otherwise ordinary-looking functions genuinely have no expression in terms of standard elementary functions at all, a fact that can be proven, not just a sign that the right technique simply hasn't been found yet.
What we're still unsure about
That differentiation admits a complete, mechanical algorithm while integration does not is a rigorously proven mathematical fact, not an open question — certain classes of functions are formally proven to have no elementary antiderivative, meaning no amount of cleverness with standard techniques will ever find one. What's more a matter of ongoing mathematical and computational work, rather than settled theory, is exactly how far automated symbolic integration software can be pushed to reliably recognise which technique, or combination of techniques, will solve a given solvable integral — modern computer algebra systems handle an impressively wide range of integrals automatically, but they still rely on large libraries of pattern-matching heuristics rather than a single, universally guaranteed procedure equivalent to what exists for differentiation.
This sits inside Techniques of Integration, one of eight topics in Calculus, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.