Learn & Review: Calculus 2 - Basic Integration

Calculus II Integration Techniques Summary

This video covers several key integration techniques taught in Calculus II, including integration by parts, trigonometric integrals, and trigonometric substitution.

1. Integration by Parts

This technique is used when integrating a product of two different types of functions.

• Formula: ∫ u dv = uv - ∫ v du

• Process:

  1. Identify u and dv from the integrand. A common mnemonic is LIATE (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) to help choose u.
  2. Calculate du by differentiating u.
  3. Calculate v by integrating dv.
  4. Substitute these into the integration by parts formula.
  5. Simplify and solve the resulting integral.

• Example 1: ∫ x sin(x) dx

  • Let u = x (algebraic) and dv = sin(x) dx (trigonometric).
  • Then du = dx and v = -cos(x).
  • Applying the formula: x(-cos(x)) - ∫ (-cos(x)) dx
  • Simplifies to: -x cos(x) + ∫ cos(x) dx
  • Result: -x cos(x) + sin(x) + C

• Example 2: ∫ ln(x) dx

  • Let u = ln(x) and dv = dx.
  • Then du = (1/x) dx and v = x.
  • Applying the formula: x ln(x) - ∫ x * (1/x) dx
  • Simplifies to: x ln(x) - ∫ 1 dx
  • Result: x ln(x) - x + C

2. Trigonometric Integrals

These involve integrating powers of trigonometric functions.

• Strategy for Integrals of the form ∫ cosⁿ(x) sinᵐ(x) dx:

  • If n is odd: Save one cos(x) and convert the remaining even power of cos(x) to sin(x) using cos²(x) = 1 - sin²(x). Then use u-substitution with u = sin(x).
    • Example: ∫ cos³(x) dx
      • Rewrite as: ∫ cos²(x) cos(x) dx
      • Use identity: ∫ (1 - sin²(x)) cos(x) dx
      • Let u = sin(x), du = cos(x) dx.
      • Integral becomes: ∫ (1 - u²) du
      • Result: u - (u³/3) + C which is sin(x) - (1/3)sin³(x) + C.
  • If m is odd: Save one sin(x) and convert the remaining even power of sin(x) to cos(x) using sin²(x) = 1 - cos²(x). Then use u-substitution with u = cos(x).
  • If both n and m are even: Use power-reducing identities:
    • sin²(x) = (1/2)(1 - cos(2x))
    • cos²(x) = (1/2)(1 + cos(2x))
    • Example: ∫ sin²(x) dx
      • Use identity: ∫ (1/2)(1 - cos(2x)) dx
      • Integrate term by term: (1/2)x - (1/2) * (1/2)sin(2x) + C
      • Result: (1/2)x - (1/4)sin(2x) + C

• Integrals involving tangent and secant (∫ tanⁿ(x) secᵐ(x) dx):

  • If m is even: Save sec²(x) and convert the remaining even power of sec(x) to tan(x) using sec²(x) = 1 + tan²(x). Then use u-substitution with u = tan(x).
    • Example: ∫ tan⁶(x) sec⁴(x) dx
      • Rewrite as: ∫ tan⁶(x) sec²(x) sec²(x) dx
      • Use identity: ∫ tan⁶(x) (1 + tan²(x)) sec²(x) dx
      • Let u = tan(x), du = sec²(x) dx.
      • Integral becomes: ∫ u⁶(1 + u²) du = ∫ (u⁶ + u⁸) du
      • Result: (u⁷/7) + (u⁹/9) + C which is (1/7)tan⁷(x) + (1/9)tan⁹(x) + C.
  • If n is odd: Save sec(x)tan(x) and convert the remaining even power of tan(x) to sec(x) using tan²(x) = sec²(x) - 1. Then use u-substitution with u = sec(x).

3. Trigonometric Substitution

This technique is used for integrals involving expressions of the form √(a² - x²), √(a² + x²), or √(x² - a²).

• Substitution Rules:

  • For √(a² - x²): Let x = a sin(θ)
  • For √(a² + x²): Let x = a tan(θ)
  • For √(x² - a²): Let x = a sec(θ)

• Process:

  1. Identify the form of the expression and choose the appropriate substitution.
  2. Calculate dx in terms of dθ.
  3. Substitute x and dx into the integral.
  4. Simplify the expression, often using trigonometric identities (e.g., 1 - sin²(θ) = cos²(θ), 1 + tan²(θ) = sec²(θ), sec²(θ) - 1 = tan²(θ)).
  5. Integrate with respect to θ.
  6. Convert the result back to x using a right triangle based on the initial substitution.

• Example: ∫ √(4 - x²) / x² dx

  • Form is √(a² - x²), with a = 2.
  • Substitution: x = 2 sin(θ), so dx = 2 cos(θ) dθ.
  • √(4 - x²) = √(4 - 4 sin²(θ)) = √(4 cos²(θ)) = 2 cos(θ).
  • Integral becomes: (2 cos(θ)) / (4 sin²(θ)) * (2 cos(θ) dθ)
  • Simplifies to: (4 cos²(θ)) / (4 sin²(θ)) dθ = cot²(θ) dθ.
  • Use identity: cot²(θ) = csc²(θ) - 1.
  • Integral becomes: ∫ (csc²(θ) - 1) dθ.
  • Integrate: -cot(θ) - θ + C.
  • Convert back to x:
    • From x = 2 sin(θ), we get sin(θ) = x/2.
    • Construct a right triangle: opposite = x, hypotenuse = 2, adjacent = √(4 - x²).
    • cot(θ) = adjacent / opposite = √(4 - x²) / x.
    • θ = arcsin(x/2).
  • Result: -√(4 - x²) / x - arcsin(x/2) + C.