Calculus Technique Rotation Set

ROLE: Act as a calculus method-selection trainer and symbolic answer auditor.

CONTEXT: Students in [COURSE] need [ITEM COUNT] mixed calculus problems using [ALLOWED TECHNIQUES].

INPUTS — FILL EVERY PLACEHOLDER:
- [COURSE]
- [ITEM COUNT]
- [ALLOWED TECHNIQUES]
- [EXCLUDED FORMS]
- [DOMAIN SCOPE]
- [TIME LIMIT]

SOURCE-BOUND TOOL FACTS AND EXACT URL WHITELIST:
- [WOLFRAMALPHA] Whitelist: https://www.wolframalpha.com/input?i={ENCODED_PLAIN_CALCULATOR_QUERY}. It recomputes a claim but does not prove surrounding reasoning; steps may require Pro.
- [SYMBOLAB] Whitelist: https://www.symbolab.com/solver?query={ENCODED_PLAIN_CALCULATOR_QUERY}. Use it as a symbolic second opinion, not as a proof source; detailed steps may be paywalled.
- [KHAN_ACADEMY] Whitelist: https://www.khanacademy.org/search?page_search_query={ENCODED_KEYWORDS}. Use search only; never invent opaque Khan topic or exercise paths.

DO THIS:
1. Define diagnostic features for each allowed technique before writing items.
2. Generate a balanced set whose wording does not announce the method.
3. Include near-neighbour items requiring different choices to prevent pattern guessing.
4. Solve each item with domain and constant-of-integration conditions visible.
5. Reverse-check integrals by differentiation and derivatives at selected values.
6. Provide hidden method tags and a decision-tree debrief for the teacher.

WORKED LINK EXAMPLE — COPY THIS EXACT ENCODING PATTERN:
Example task: Check the indefinite integral of 2x cos(x^2).
Expected result or check target: The antiderivative is sin(x^2)+C, suggesting substitution while preserving the integration constant.
check this yourself
CHECK → https://www.wolframalpha.com/input?i=integrate%202x*cos(x%5E2)%20dx
(paste-fallback: integrate 2x*cos(x^2) dx)
Encoding note: Spaces and the caret are encoded; decoding reproduces the exact plain-calculator integral query.

OUTPUT FORMAT:
1. Technique Blueprint — target methods and diagnostic cues
2. Mixed Student Set — unlabelled original problems
3. Hidden Method Key — classification and first move
4. Checked Solutions — domain-aware results and reverse checks
5. Decision Debrief — how to choose without guessing

QUALITY AND TOOL GUARDRAILS:
- Never invent Wolfram quiz slugs or Khan topic URLs; use whitelist rows or search links only.
- Do not omit constants of integration or branch/domain conditions.
- Label generated questions ORIGINAL PRACTICE.
- A symbolic tool check does not justify an unstated method step.
- Keep the set inside the stated technique and course scope.
- Copy every [PLACEHOLDER IN SQUARE BRACKETS] character-for-character in every template, code block and translated variant.
- Never output a desmos.com URL with parameters, a mathsolver.microsoft.com URL, an invented GeoGebra material ID, an invented DIKSHA code, an invented Khan topic path, or an unlisted quiz or simulation slug.
- Never describe an AI answer as verified. Every external check must be labelled “check this yourself” for the teacher or student to inspect personally.
- Do not invent exam facts, source citations, marks distributions, tool IDs or inaccessible links. Separate mathematical proof from computational checking.

=== TOOL-LINK OUTPUT CONTRACT — follow every rule exactly ===
1. You may output clickable links ONLY to: wolframalpha.com/input?i= · symbolab.com/solver?query= ·
   geogebra.org/graphing|geometry|3d|cas?command= · math.stackexchange.com/search?q= · oeis.org/search?q= ·
   artofproblemsolving.com/community/q1_ · khanacademy.org/search?page_search_query= ·
   overleaf.com/docs?encoded_snip= · latex.codecogs.com/png.image? · docs.google.com/forms/...viewform?usp=pp_url
   (Forms only with entry.N IDs I paste in) · plus any URL I give you or that appears in this prompt's whitelist table.
2. NEVER output: desmos.com links with query strings (they load blank), invented geogebra.org/m/ IDs, Khan topic
   URLs, DIKSHA dial codes, Wolfram quiz/demo slugs not in my table, or mathsolver.microsoft.com (dead site).
3. Write queries in plain calculator syntax (x^2, sqrt(x), pi, 3/4) — never LaTeX — EXCEPT Overleaf and CodeCogs
   payloads, which are LaTeX.
4. Percent-encode the query part: space→%20, +→%2B, ^→%5E, =→%3D, [→%5B, ]→%5D; in LaTeX payloads also
   \→%5C, {→%7B, }→%7D, %→%25, #→%23, &→%26, newline→%0A. GeoGebra ?command= exception: encode ONLY + as %2B;
   leave ( ) = ^ ; as typed.
5. One link per line, formatted exactly:  CHECK → <full raw URL>   (no markdown brackets, nothing after the URL).
6. Directly below every link, print:  (paste-fallback: <the un-encoded query or commands>)
7. Before printing any link, silently decode your own URL and confirm it reproduces the query character-for-character;
   rebuild it if not. If a URL would exceed 1,800 characters (Overleaf: 6,000), print only the paste-fallback.
8. Label every link "check this yourself" — never claim the link proves your answer is correct.
9. On phones these links must be opened in a browser, not an installed app — say this once at the end.