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UNIT 6 · 17–20%

Integration & Accumulation

from little rectangles to the Fundamental Theorem ∫

📦 antiderivatives

F is an antiderivative of f if F′(x) = f(x). The general antiderivative includes +C unless initial conditions pin it down.

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)  ·  ∫ 1/x dx = ln|x| + C

📐 Riemann sums & definite integral

Left/right/midpoint/trapezoidal sums approximate ∫ab f(x) dx using rectangles or trapezoids. As partitions refine (n→∞), sums approach the definite integral when f is integrable.

∫ab f(x) dx = accumulated signed area under y = f(x) from a to b

🏆 Fundamental Theorem of Calculus

If F′ = f, then ∫ab f(x) dx = F(b) − F(a)
If g(x) = ∫ax f(t) dt and f continuous, then g′(x) = f(x)

Watch variable limits (chain rule!) when bounds are functions of x.

🔄 u-substitution

Reverse chain rule: pick u so that du absorbs extra factors (or solves for dx). Change limits in definite integrals or substitute back.

🧩 integration by parts BC

∫ u dv = uv − ∫ v du

LIATE helps choose u: Log, Inverse trig, Algebraic, Trig, Exponential — earlier in list often better as u.

✏️ quick practice

  1. Evaluate ∫ (2x + 1)³ dx using substitution.
  2. Find d/dx [∫0x² sin(t²) dt].
  3. Compute ∫ x eˣ dx using integration by parts.
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slope fields & DEs next 〰️