Evaluation of Definite and Indefinite Integrals

1. Understanding Integration

Integration is the reverse process of differentiation and is used to find areas under curves, accumulated quantities, and solve differential equations. There are two main types of integrals:

• Indefinite Integral: Represents a family of functions and includes a constant of integration (C).
• Definite Integral: Represents the net area under the curve between two bounds and yields a numerical value.

2. Basic Integration Rules

• ∫ xⁿ dx = (xⁿ⁺¹)⁄(n+1) + C for n ≠ -1
• ∫ (1/x) dx = ln|x| + C
• ∫ eˣ dx = eˣ + C
• ∫ sin x dx = -cos x + C
• ∫ cos x dx = sin x + C

3. Techniques of Integration

Some problems require techniques beyond the basic rules:

• Substitution: Useful when the integrand includes a composite function.
• Integration by Parts: Based on the product rule: ∫ u·dv = uv - ∫ v·du

4. Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration. It states:

• If F is an antiderivative of f, then ∫_a^b f(x) dx = F(b) - F(a)

5. Example Problem

Problem: Evaluate ∫₁³ (2x + 1) dx

1. Find the antiderivative: ∫ (2x + 1) dx = x² + x + C
2. Apply the limits: [x² + x]₁³ = (9 + 3) - (1 + 1) = 12 - 2 = 10
3. Answer: 10