Fundamental Integrals

Unit I: Fundamental Integrals

1.1 Introduction to Integration

Integration is the reverse process of differentiation. While differentiation finds the rate of change of a function, integration helps us find the original function when its derivative is known. This fundamental concept appears throughout mathematics, physics, engineering, and economics.

There are two main types of integrals:

• Indefinite Integrals - These represent families of functions and include an arbitrary constant
• Definite Integrals - These calculate the signed area under a curve between two specific points

In integral notation, ∫f(x)dx represents the integral of function f(x) with respect to x.

1.2 Indefinite Integrals

An indefinite integral is written as:

∫f(x)dx = F(x) + C

where:

• F(x) is the antiderivative (the original function)
• C is the constant of integration (arbitrary constant)
• f(x) is the integrand (the function being integrated)

Standard Integration Formulas:

1.3 Integration by Substitution

This technique (also called u-substitution) is used when the integrand contains a composite function. The method involves:

1. Identify a function u inside the composite function
2. Calculate du = u'(x)dx
3. Rewrite the integral in terms of u
4. Integrate with respect to u
5. Substitute back the original variable

Example: Evaluate ∫2x·cos(x²)dx

Let u = x², then du = 2x dx
The integral becomes: ∫cos(u)du = sin(u) + C
Substituting back: sin(x²) + C

1.4 Integration by Parts

This technique is useful when the integrand is a product of two functions. The formula is:

∫u dv = uv - ∫v du

Strategy: Use LIATE rule to choose u (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential functions in order of preference).

Example: Evaluate ∫x·e^xdx

Let u = x, dv = e^xdx
Then du = dx, v = e^x
Using the formula: ∫x·e^xdx = x·e^x - ∫e^xdx = x·e^x - e^x + C = e^x(x - 1) + C

1.5 Integration by Partial Fractions

This method decomposes rational functions (fractions with polynomials) into simpler parts that are easier to integrate.

Steps:

1. Ensure the degree of numerator < degree of denominator (if not, perform polynomial division first)
2. Factor the denominator completely
3. Write partial fraction decomposition using appropriate forms
4. Solve for constants
5. Integrate each partial fraction

Example: Evaluate ∫(5x + 1)/(x² + 2x - 3)dx

Factor: x² + 2x - 3 = (x + 3)(x - 1)
Write: (5x + 1)/[(x + 3)(x - 1)] = A/(x + 3) + B/(x - 1)
Solving: A = 2, B = 3
∫(5x + 1)/(x² + 2x - 3)dx = 2ln|x + 3| + 3ln|x - 1| + C

1.6 Definite Integrals

A definite integral calculates the net signed area under a curve between two points:

∫_a^bf(x)dx = F(b) - F(a)

where F is an antiderivative of f, and [a, b] is the interval of integration.

Fundamental Theorem of Calculus (Part 2):

This theorem establishes the relationship between differentiation and integration. If f is continuous on [a, b] and F is an antiderivative of f, then the definite integral equals the difference of F evaluated at the endpoints.

Example: Evaluate ∫₀²x²dx

Find antiderivative: F(x) = x³/3
Apply limits: F(2) - F(0) = 8/3 - 0 = 8/3

1.7 Improper Integrals

Improper integrals involve:

• Infinite limits of integration (∫_a^∞ or ∫_-∞^b)
• Discontinuous integrands within the interval

For improper integrals with infinite limits:

∫_a^∞f(x)dx = lim_t→∞ ∫_a^tf(x)dx

Example: Evaluate ∫₁^∞1/x²dx

= lim_t→∞ ∫₁^tx^-2dx
= lim_t→∞ [-1/x]₁^t
= lim_t→∞ (-1/t + 1) = 1

1.8 Beta and Gamma Functions

Gamma Function: Extends the factorial function to real and complex numbers.

Γ(n) = ∫₀^∞x^n-1e^-xdx

Property: Γ(n) = (n-1)! for positive integers n

Beta Function: Related to the Gamma function.

B(m, n) = ∫₀¹x^m-1(1-x)^n-1dx = Γ(m)Γ(n)/Γ(m+n)

1.9 Double Integrals (Concept)

Double integrals extend single integrals to functions of two variables f(x, y):

∫∫_Rf(x,y) dA

This represents the volume under a 3D surface over a region R in the xy-plane. Evaluated iteratively using Fubini's theorem:

∫_a^b∫_c^df(x,y) dy dx

Integration is performed first with respect to y (treating x as constant), then with respect to x.