Indefinite Integrals

• Integral calculus is a branch of calculus concerned with the accumulation of quantities and the areas under or between curves
• It is the inverse operation to differential calculus
• The process of finding an integral is called integration

Indefinite Integrals

• An indefinite integral is a function that represents the antiderivative of another function
• It expresses the most general function whose derivative is the original function
• Notation: ∫ f(x) dx = F(x) + C, where F'(x) = f(x) and C is the constant of integration
• The constant of integration, C, signifies that there are infinitely many antiderivatives of a function differing only by a constant
• Basic power rule: ∫ x^n dx = (x^(n+1))/(n+1) + C, for n ≠ -1
• Integral of 1/x: ∫ (1/x) dx = ln|x| + C
• Integral of e^x: ∫ e^x dx = e^x + C
• Integral of a^x: ∫ a^x dx = (a^x)/ln(a) + C
• Integral of sin(x): ∫ sin(x) dx = -cos(x) + C
• Integral of cos(x): ∫ cos(x) dx = sin(x) + C
• Integral of sec^2(x): ∫ sec^2(x) dx = tan(x) + C
• Integral of csc^2(x): ∫ csc^2(x) dx = -cot(x) + C
• Integral of sec(x)tan(x): ∫ sec(x)tan(x) dx = sec(x) + C
• Integral of csc(x)cot(x): ∫ csc(x)cot(x) dx = -csc(x) + C
• Integral of 1/(sqrt(1-x^2)): ∫ 1/(√(1-x^2)) dx = arcsin(x) + C
• Integral of -1/(sqrt(1-x^2)): ∫ -1/(√(1-x^2)) dx = arccos(x) + C
• Integral of 1/(1+x^2): ∫ 1/(1+x^2) dx = arctan(x) + C

Definite Integrals

• A definite integral is an integral with upper and lower limits of integration
• It calculates the signed area under a curve between these limits
• Notation: ∫[a to b] f(x) dx = F(b) - F(a), where F(x) is an antiderivative of f(x)
• Fundamental Theorem of Calculus, Part 1: d/dx ∫[a to x] f(t) dt = f(x)
• Fundamental Theorem of Calculus, Part 2: ∫[a to b] f(x) dx = F(b) - F(a), where F'(x) = f(x)
• Properties of Definite Integrals:
  • ∫[a to a] f(x) dx = 0
  • ∫[a to b] f(x) dx = -∫[b to a] f(x) dx
  • ∫[a to b] cf(x) dx = c∫[a to b] f(x) dx, where c is a constant
  • ∫[a to b] [f(x) + g(x)] dx = ∫[a to b] f(x) dx + ∫[a to b] g(x) dx
  • ∫[a to c] f(x) dx + ∫[c to b] f(x) dx = ∫[a to b] f(x) dx

Integration Techniques

• Substitution (u-substitution):
  • Used to simplify integrals by substituting a function with a new variable, u
  • Steps: Choose u = g(x), find du = g'(x) dx, substitute u and du into the integral, evaluate the new integral with respect to u, substitute back g(x) for u
• Integration by Parts:
  • Used to integrate products of functions
  • Formula: ∫ u dv = uv - ∫ v du
  • Choose u and dv appropriately using guidelines like LIATE (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) to make the integral simpler
• Trigonometric Integrals:
  • Involve trigonometric functions
  • Use trigonometric identities to simplify the integrand
  • ∫ sin^m(x) cos^n(x) dx: If m is odd, save a sin(x) and use sin^2(x) = 1 - cos^2(x); if n is odd, save a cos(x) and use cos^2(x) = 1 - sin^2(x); if both are even, use half-angle formulas
  • ∫ tan^m(x) sec^n(x) dx: If n is even, save a sec^2(x) and use sec^2(x) = 1 + tan^2(x); if m is odd, save a sec(x)tan(x) and use tan^2(x) = sec^2(x) - 1
• Trigonometric Substitution:
  • Used for integrals containing √(a^2 - x^2), √(a^2 + x^2), or √(x^2 - a^2)
  • For √(a^2 - x^2), use x = a sin(θ)
  • For √(a^2 + x^2), use x = a tan(θ)
  • For √(x^2 - a^2), use x = a sec(θ)
• Partial Fractions:
  • Used to integrate rational functions (polynomial/polynomial)
  • Decompose the rational function into simpler fractions
  • Applicable when the degree of the numerator is less than the degree of the denominator
  • Factor the denominator and express the rational function as a sum of fractions with unknown constants in the numerators
  • Solve for the unknown constants
• Improper Integrals:
  • Integrals where either the interval of integration is infinite or the function has a vertical asymptote within the interval of integration
  • ∫[a to ∞] f(x) dx = lim (t→∞) ∫[a to t] f(x) dx
  • ∫[-∞ to b] f(x) dx = lim (t→-∞) ∫[t to b] f(x) dx
  • ∫[-∞ to ∞] f(x) dx = ∫[-∞ to c] f(x) dx + ∫[c to ∞] f(x) dx (evaluate each separately)
  • If f(x) has a vertical asymptote at x = c in [a, b], then ∫[a to b] f(x) dx = lim (t→c-) ∫[a to t] f(x) dx + lim (t→c+) ∫[t to b] f(x) dx

Applications of Integration

• Area between curves: ∫[a to b] |f(x) - g(x)| dx, where f(x) and g(x) are the curves
• Average value of a function: (1/(b-a)) ∫[a to b] f(x) dx
• Volume of solid of revolution:
  • Disk method: V = π ∫[a to b] [f(x)]^2 dx (rotation about x-axis) or V = π ∫[c to d] [f(y)]^2 dy (rotation about y-axis)
  • Washer method: V = π ∫[a to b] ([f(x)]^2 - [g(x)]^2) dx (rotation about x-axis)
  • Shell method: V = 2π ∫[a to b] x f(x) dx (rotation about y-axis)
• Arc length: ∫[a to b] √(1 + [f'(x)]^2) dx
• Surface area of revolution:
  • Rotation about x-axis: 2π ∫[a to b] f(x) √(1 + [f'(x)]^2) dx
  • Rotation about y-axis: 2π ∫[a to b] x √(1 + [f'(x)]^2) dx