Unit 5: Integration II

Integration extends the anti-derivative of Unit I into two directions: a stronger technique for products of functions, and the definite integral that attaches a number to a curve over an interval. The definite integral is defined rigorously as a limit of sums and then linked to anti-derivatives by the Fundamental Theorem of Calculus.

• Indefinite integral: the family of anti-derivatives, written ∫f(x) dx = F(x) + C, where F'(x) = f(x) and C is arbitrary.
• Definite integral: a single real number ∫ₐᵇ f(x) dx fixed once the limits a (lower) and b (upper) are chosen; no constant C survives.
• Continuity assumption: every function integrated below is assumed continuous, or at worst bounded with finitely many jumps, on the closed interval [a, b].
• Fundamental Theorem of Calculus (FTC): if F is an anti-derivative of continuous f, then ∫ₐᵇ f(x) dx = F(b) − F(a), written [F(x)]ₐᵇ.
• Sign convention: area above the x-axis counts positive, area below counts negative; the integral gives the net signed area.

II. Integration by Parts

Reversing the product rule to integrate a product of two functions.

A. Statement and principle

The method integrates ∫u·v dx when the integrand is a product whose two factors behave differently — one simplifies on differentiation, the other stays integrable.

• Formula: derived from (uv)' = u'v + uv', rearranged and integrated:
  
  TEXT

  ∫ u · v dx = u ∫v dx − ∫ [ (du/dx) · ∫v dx ] dx

• Symbols: u is the factor chosen to be differentiated, v the factor to be integrated; du/dx is the derivative of u, and ∫v dx the anti-derivative of v.
• Core idea: the new integral ∫(du/dx)(∫v dx) dx must be simpler than the original, otherwise the choice of u and v was wrong.

B. Choosing the first function (ILATE rule)

The order of preference for which factor becomes u follows the mnemonic ILATE.

• I — Inverse trig: e.g. sin⁻¹x, chosen first because it differentiates to an algebraic form.
• L — Logarithmic: e.g. log x, which has no elementary anti-derivative but a simple derivative 1/x.
• A — Algebraic: e.g. x², powers that reduce in degree on differentiation.
• T — Trigonometric: e.g. sin x, cos x.
• E — Exponential: e.g. eˣ, taken as v last because it integrates cleanly.

C. Worked example and special cases

Take ∫ x·eˣ dx, an Algebraic × Exponential product.

• Assignment: u = x (A before E), v = eˣ, so du/dx = 1, ∫v dx = eˣ.
• Apply: x·eˣ − ∫ eˣ·1 dx = x·eˣ − eˣ + C = eˣ(x − 1) + C.
• Single-function trick: ∫ log x dx is treated as ∫ (log x)·1 dx with v = 1, giving x log x − x + C.
• Standard result: ∫ eˣ[f(x) + f'(x)] dx = eˣ·f(x) + C, a direct consequence useful for integrands of that shape.

III. The Definite Integral

Attaching a fixed numerical value to the integral over a closed interval.

A. Definition via the Fundamental Theorem

The definite integral evaluates the anti-derivative at the two limits and subtracts.

• Rule: ∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a), where F'(x) = f(x).
• Constant cancels: using F(x) + C, the +C at both ends subtracts away, so the answer is independent of the choice of anti-derivative.
• Geometric meaning: for f(x) ≥ 0, the value equals the area bounded by the curve y = f(x), the x-axis, and the lines x = a and x = b.
• Example: ∫₀¹ x² dx = [x³/3]₀¹ = 1/3 − 0 = 1/3.

B. Evaluating with substitution

When a substitution changes the variable, the limits must change with it.

• Method: if t = g(x), replace dx and rewrite the limits as t = g(a) and t = g(b); there is then no need to revert to x.
• Example: ∫₀¹ 2x·e^{x²} dx. Put t = x², so dt = 2x dx; limits become t = 0 to t = 1, giving ∫₀¹ eᵗ dt = [eᵗ]₀¹ = e − 1.

IV. Definite Integral as the Limit of a Sum

Defining the integral from first principles as an infinite sum of thin rectangles.

A. Construction of the sum

The interval is sliced into equal strips whose areas are added and refined to a limit.

• Partition: divide [a, b] into n equal subintervals of width h = (b − a)/n.
• Sample points: heights are read at a, a+h, a+2h, …, a+(n−1)h, one per strip.
• Riemann sum: the total rectangle area is
  
  TEXT

  Sₙ = h [ f(a) + f(a+h) + f(a+2h) + … + f(a+(n−1)h) ]

• Definition: the definite integral is the limit as strips become infinitely thin:
  
  TEXT

  ∫ₐᵇ f(x) dx = lim (n→∞) h · Σ (r=0 to n−1) f(a + r·h),   h = (b−a)/n

• Symbols: n is the number of strips, h their common width, r the strip index; as n→∞, h→0.

B. Evaluation from the definition

The limit is computed using standard summation formulae rather than an anti-derivative.

• Needed sums: Σr = n(n−1)/2, Σr² = (n−1)n(2n−1)/6, and Σ1 = n over r = 0 to n−1.
• Example: evaluate ∫₀¹ x dx from first principles. Here a = 0, b = 1, h = 1/n, f(x) = x:
  
  TEXT

  Sₙ = h · Σ (r=0 to n−1) (r·h) = h² · Σr = (1/n²)·[n(n−1)/2]
     = (n−1)/(2n) = 1/2 − 1/(2n)

• Take the limit: as n→∞, 1/(2n)→0, so the integral is 1/2, matching [x²/2]₀¹.
• Purpose: this confirms that the FTC value and the geometric area agree, grounding the shortcut in a genuine definition.

V. Some Properties of Definite Integrals

Structural rules that simplify evaluation before any anti-derivative is found.

A. Limit and interval properties

These govern how the value responds to the limits of integration.

• Reversal: ∫ₐᵇ f(x) dx = − ∫ᵇₐ f(x) dx; swapping limits flips the sign.
• Zero width: ∫ₐᵃ f(x) dx = 0, since there is no interval.
• Additivity (splitting): ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx for any c, typically a < c < b; useful for piecewise functions.
• Variable is dummy: ∫ₐᵇ f(x) dx = ∫ₐᵇ f(t) dt; the name of the integration variable does not affect the value.

B. Reflection and symmetry properties

These exploit how the integrand behaves under a change of variable.

• King's rule: ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b − x) dx, replacing x by a+b−x; it often turns an awkward integrand into a solvable one.
• Zero-to-a form: ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx, the special case with a = 0 as lower limit.
• Symmetric interval: over [−a, a],
  
  TEXT

  ∫₋ₐᵃ f(x) dx = 2 ∫₀ᵃ f(x) dx   if f is even, f(−x) = f(x)
  ∫₋ₐᵃ f(x) dx = 0               if f is odd, f(−x) = −f(x)

• Example: ∫₋₁¹ x³ dx = 0, since x³ is odd; no computation of the anti-derivative is needed.

C. Order and periodicity properties

These bound the value and handle repeating functions.

• Doubling over a period: ∫₀^{2a} f(x) dx = 2∫₀ᵃ f(x) dx when f(2a − x) = f(x), and equals 0 when f(2a − x) = −f(x).
• Periodic functions: ∫₀^{nT} f(x) dx = n ∫₀ᵀ f(x) dx when f has period T; each period contributes the same area.
• Order preservation: if f(x) ≤ g(x) on [a, b], then ∫ₐᵇ f(x) dx ≤ ∫ₐᵇ g(x) dx; integration keeps inequalities.
• Sign rule: if f(x) ≥ 0 throughout [a, b], then ∫ₐᵇ f(x) dx ≥ 0; the signed area cannot be negative when the curve stays above the axis.

D. Application of King's rule

The reflection property collapses certain integrals by adding the integral to its mirror image.

• Standard result: ∫₀^{π/2} [sinⁿx / (sinⁿx + cosⁿx)] dx = π/4 for any n.
• Reasoning: call the integral I; applying x → π/2 − x swaps sin and cos, giving a second expression I'. Adding I + I' = ∫₀^{π/2} 1 dx = π/2, and since I = I', each equals π/4.
• Value: the technique returns a closed number without integrating the messy quotient at all.