4️⃣ TRIG INTEGRALS • Powers of sin/cos: - If odd power of sin → u = cos. - If odd power of cos → u = sin. • Both even → use identities: sin²x = (1‑cos2x)/2, cos²x = (1+cos2x)/2 • Products sin·cos → use substitution u = sin or cos.
Tip: After
2️⃣ INTEGRATION BY PARTS • Formula: ∫u dv = uv – ∫v du • Choose u = “algebraic” (poly, log) → du simpler. • dv = “trig, exp, power” → v easy. integral calculus by das and mukherjee pdf
6️⃣ COMMON INTEGRALS (keep this table) ∫dx/(x²+a²) = (1/a) arctan(x/a)+C ∫dx/(x²–a²) = (1/2a) ln| (x‑a)/(x+a) |+C ∫dx/√(a²‑x²) = arcsin(x/a)+C ∫dx/√(x²‑a²) = ln|x+√(x²‑a²)|+C 4️⃣ TRIG INTEGRALS • Powers of sin/cos: -
------------------------------------------------- Print and stick it on the inside cover of your notebook. Use it as a first‑pass reference each time you start a new problem. | Week | Focus | Goal (hours) | Sample Problems | |------|-------|--------------|-----------------| | 1 | Definite integrals & basic techniques | 8–10 hrs | Compute area between curves, evaluate ∫₀^π/2 sin²x dx, ∫₁^4 (x³‑2x)dx | | 2 | Advanced techniques (partial fractions, trig subs) | 10–12 hrs | ∫(2x+3)/(x²‑x‑6)dx, ∫dx/√(9‑x²), ∫tan³x sec²x dx | | 3 | Applications (volumes, arc length, work) | 12 hrs | Volume of solid generated by rotating y = √x about x‑axis, surface area of y = ln x, work done pulling a rope | | 4 | Improper integrals & ODEs | 10 hrs | Test convergence of ∫₁^∞ 1/(x ln²x)dx, solve dy/dx = y·tan x, find particular solution with y(0)=2 | Tip: After 2️⃣ INTEGRATION BY PARTS • Formula:
7️⃣ QUICK CHECKLIST BEFORE FINISHING • Did I revert u→x (if indefinite)? • Did I adjust limits (if definite)? • Is the result simplified? (Combine logs, rationalize) • Add constant C for indefinite integrals.