14. INTEGRAL CALCULUS

Integral calculus deals with finding the anti-derivative or reverse process of differentiation. The process of finding an integral is called integration. It is used to calculate areas, volumes, and accumulated quantities in science and engineering.

Basic Concept

If F′(x) = f(x), then:

∫ f(x) dx = F(x) + C

C is the constant of integration.

Standard Integrals

  • ∫ dx = x + C
  • ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1
  • ∫ 1/x dx = ln|x| + C
  • ∫ eˣ dx = eˣ + C
  • ∫ aˣ dx = aˣ / ln(a) + C
  • ∫ sin x dx = −cos x + C
  • ∫ cos x dx = sin x + C
  • ∫ sec² x dx = tan x + C
  • ∫ csc² x dx = −cot x + C
  • ∫ sec x tan x dx = sec x + C
  • ∫ csc x cot x dx = −csc x + C

General Rules of Integration

1. Constant Multiple Rule

∫ k f(x) dx = k ∫ f(x) dx

2. Sum Rule

∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx

Method of Substitution

Used when expression becomes simpler by substituting u = g(x).

Steps:

  1. Let u = g(x)
  2. Compute du = g′(x) dx
  3. Rewrite integral in terms of u
  4. Integrate and back-substitute

Example (Concept Idea)

To integrate (2x) / (x² + 1): Use substitution u = x² + 1.

Integration of Rational Functions

Case 1: 1/(ax + b)

∫ 1/(ax + b) dx = (1/a) ln|ax + b| + C

Case 2: Using Partial Fractions

Useful when denominator is factorisable:

Example types:

  • 1 / (x + a)(x + b)
  • (px + q) / (quadratic)

Break expression into simpler fractions and integrate separately.

Integration by Parts

Used when integrand is a product of two functions.

Formula

∫ u dv = uv − ∫ v du

Choosing u and dv

Use the rule ILATE: Inverse, Log, Algebraic, Trigonometric, Exponential

Example (Concept Idea)

To integrate x·eˣ, take u = x and dv = eˣ dx.

Integration of Trigonometric Functions

  • ∫ tan x dx = ln|sec x| + C
  • ∫ cot x dx = ln|sin x| + C
  • ∫ sec x dx = ln|sec x + tan x| + C
  • ∫ csc x dx = ln|csc x − cot x| + C

Integration of Exponential and Logarithmic Functions

  • ∫ eᵏˣ dx = (1/k) eᵏˣ + C
  • ∫ x eˣ dx = (x − 1)eˣ + C
  • ∫ log x dx = x(log x − 1) + C

Definite Integrals

If F(x) is anti-derivative of f(x):

∫ₐᵇ f(x) dx = F(b) − F(a)

Properties of Definite Integrals

  • ∫ₐᵇ f(x) dx = − ∫ᵇₐ f(x) dx
  • ∫ₐᵃ f(x) dx = 0
  • ∫ₐᵇ [f(x) + g(x)] dx = ∫ₐᵇ f(x) dx + ∫ₐᵇ g(x) dx
  • ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx
  • ∫₋ₐᵃ f(x) dx = 2 ∫₀ᵃ f(x) dx if f(x) is even
  • ∫₋ₐᵃ f(x) dx = 0 if f(x) is odd

Geometrical Meaning of Definite Integral

The definite integral represents the area under a curve between limits x = a and x = b.

Improper Integrals (Intro Idea)

Integrals with infinite limits or where the function becomes unbounded.

Applications of Integration

  • Area under curves
  • Volume of solids of revolution
  • Distance, velocity and acceleration
  • Work done in physics
  • Concentration–time relationships in pharmacy

Examples (General Overview)

  • Integrating algebraic expressions using substitution
  • Using partial fractions for rational functions
  • Applying integration by parts to exponential–algebraic products
  • Evaluating definite integrals with limits

Practice Questions

  • Evaluate ∫ (3x² + 4x) dx
  • Find ∫ x / (x² + 1) dx
  • Find ∫ (x log x) dx
  • Evaluate ∫₀² (x² + 1) dx
  • Solve ∫ eˣ sin x dx using integration by parts

Detailed Notes:

For PDF style full-color notes, open the complete study material below:

Share your love