Calculus: Integration

• Integration, in calculus, is a fundamental concept related to finding the area under a curve or accumulating quantities.
• It is the reverse process of differentiation.
• Integrals are used extensively in mathematics, science, and engineering to solve problems involving areas, volumes, and rates of change.

Basic Concepts

• The integral of a function f(x) is denoted by ∫f(x) dx.
• The symbol ∫ represents the integral sign, f(x) is the integrand, and dx indicates that the integration is performed with respect to the variable x.
• The result of integration is a function F(x) such that F'(x) = f(x).
• F(x) is called the antiderivative or indefinite integral of f(x).
• Because the derivative of a constant is zero, the antiderivative is not unique, and it is always written with an arbitrary constant C, i.e., ∫f(x) dx = F(x) + C.
• There are two main types of integrals: indefinite integrals and definite integrals.

Indefinite Integrals

• An indefinite integral represents the most general antiderivative of a function.
• It is a function plus an arbitrary constant.
• Notation: ∫f(x) dx = F(x) + C, where F'(x) = f(x) and C is the constant of integration.
• Indefinite integrals represent a family of functions differing by a constant.

Definite Integrals

• A definite integral calculates the area under a curve between two specified limits.
• Notation: ∫ab f(x) dx, where a and b are the limits of integration.
• The definite integral results in a numerical value.
• It is defined as the limit of a Riemann sum.
• Geometrically, it represents the signed area between the curve y = f(x) and the x-axis from x = a to x = b.
• If f(x) > 0 on [a, b], the definite integral gives the actual area.
• If f(x) < 0 on [a, b], the definite integral gives the negative of the area.

Fundamental Theorem of Calculus

• The Fundamental Theorem of Calculus connects differentiation and integration.
• Part 1: If f is a continuous function on [a, x], then the function F(x) = ∫ax f(t) dt is continuous on [a, x] and differentiable on (a, x), and F'(x) = f(x).
• Part 2: If F is an antiderivative of f on [a, b], then ∫ab f(x) dx = F(b) - F(a).
• This theorem provides a method to evaluate definite integrals by finding an antiderivative of the integrand.

Basic Integration Formulas

• Power Rule: ∫xn dx = (xn+1)/(n+1) + C, for n ≠ -1
• Integral of 1/x: ∫(1/x) dx = ln|x| + C
• Exponential Function: ∫ex dx = ex + C
• Integral of ax: ∫ax dx = (ax)/ln(a) + C
• Sine Function: ∫sin(x) dx = -cos(x) + C
• Cosine Function: ∫cos(x) dx = sin(x) + C
• Secant Squared Function: ∫sec2(x) dx = tan(x) + C
• Cosecant Squared Function: ∫csc2(x) dx = -cot(x) + C
• Secant Tangent Function: ∫sec(x)tan(x) dx = sec(x) + C
• Cosecant Cotangent Function: ∫csc(x)cot(x) dx = -csc(x) + C

Techniques of Integration

• Substitution (u-substitution): A method used to simplify integrals by substituting a function with a new variable u.
• Integration by Parts: A technique derived from the product rule of differentiation.
• Formula: ∫u dv = uv - ∫v du
• Trigonometric Integrals: Involve trigonometric functions and often require trigonometric identities to simplify.
• Trigonometric Substitution: Used for integrals containing expressions like √(a2 - x2), √(a2 + x2), or √(x2 - a2).
• Partial Fractions: Used to integrate rational functions by decomposing them into simpler fractions.

Properties of Definite Integrals

• Integral of a Constant Multiple: ∫ab cf(x) dx = c∫ab f(x) dx
• Integral of a Sum or Difference: ∫ab [f(x) ± g(x)] dx = ∫ab f(x) dx ± ∫ab g(x) dx
• Additivity: ∫ac f(x) dx + ∫cb f(x) dx = ∫ab f(x) dx
• Reversing Limits: ∫ba f(x) dx = -∫ab f(x) dx
• If f(x) ≥ 0 on [a, b], then ∫ab f(x) dx ≥ 0.
• If f(x) ≤ 0 on [a, b], then ∫ab f(x) dx ≤ 0.
• If f(x) ≥ g(x) on [a, b], then ∫ab f(x) dx ≥ ∫ab g(x) dx.

Applications of Integration

• Area Between Curves: The area between two curves y = f(x) and y = g(x) from x = a to x = b is given by ∫ab |f(x) - g(x)| dx.
• Volume of Solids of Revolution: Found using methods like the disk method, washer method, or shell method.
• Arc Length: The length of a curve y = f(x) from x = a to x = b is given by ∫ab √(1 + [f'(x)]2) dx.
• Surface Area of Revolution: The surface area generated by revolving a curve y = f(x) from x = a to x = b about the x-axis is given by ∫ab 2πf(x)√(1 + [f'(x)]2) dx.
• Average Value of a Function: The average value of f(x) on [a, b] is given by (1/(b-a))∫ab f(x) dx.

Improper Integrals

• Integrals where one or both limits of integration are infinite or where the integrand has a vertical asymptote within the interval of integration.
• Type 1: Infinite Limits of Integration: ∫∞a f(x) dx = limt→∞ ∫ta f(x) dx, ∫b-∞ f(x) dx = limt→-∞ ∫bt f(x) dx
• Type 2: Discontinuous Integrand: If f(x) has a discontinuity at c in [a, b], then ∫ab f(x) dx = limt→c- ∫at f(x) dx + limt→c+ ∫tb f(x) dx.
• These integrals converge if the limit exists and is finite; otherwise, they diverge.