Integration for NDA 2 2025: Easy Guide to Calculus Fundamentals

Building upon the foundation of differentiation, integration is another crucial aspect of calculus for the NDA 2 2025 Mathematics paper. Integration can be thought of as the reverse process of differentiation and is used to find areas under curves, volumes, and solve various other problems. This article introduces the fundamental concepts of integration that NDA 2025 aspirants need to understand.

Fundamental Concepts of Integration:

• Indefinite Integral: The indefinite integral of a function f(x) is a function F(x) whose derivative is f(x), i.e., F′(x)=f(x). It is denoted by ∫f(x)dx=F(x)+C, where C is the constant of integration.
• Basic Integration Formulas: Mastering these formulas is essential:
  • ∫xndx=n+1xn+1​+C (where n=−1)
  • ∫x1​dx=ln∣x∣+C
  • ∫exdx=ex+C
  • ∫axdx=lnaax​+C
  • ∫sinxdx=−cosx+C
  • ∫cosxdx=sinx+C
  • ∫sec2xdx=tanx+C
  • ∫csc2xdx=−cotx+C
  • ∫secxtanxdx=secx+C
  • ∫cscxcotxdx=−cscx+C
• Definite Integral: The definite integral of a function f(x) from a to b is denoted by ∫ab​f(x)dx and represents the area under the curve y=f(x) between the limits x=a and x=b. It is calculated as ∫ab​f(x)dx=[F(x)]ab​=F(b)−F(a), where F(x) is the antiderivative of f(x).
• Properties of Definite Integrals: Understanding key properties like ∫ab​f(x)dx=−∫ba​f(x)dx, ∫ac​f(x)dx+∫cb​f(x)dx=∫ab​f(x)dx, etc.
• Basic Methods of Integration:
  • Integration by Substitution: Used when the integrand is a composite function.
  • Integration by Parts: Used for integrating the product of two functions: ∫udv=uv−∫vdu.

Applications of Integration (Basic Ideas):

• Finding the Area Under a Curve.
• Finding the Volume of Solids of Revolution (basic idea).

Preparation Strategy for Integration:

• Master Basic Formulas: Memorize the fundamental integration formulas.
• Practice Different Methods: Learn and practice integration by substitution and by parts.
• Understand Definite Integrals: Know how to evaluate definite integrals and apply their properties.
• Relate to Differentiation: Understand the inverse relationship between differentiation and integration.

Building a strong foundation in the basics of integration is essential for tackling calculus problems in the NDA 2 2025 Mathematics paper.

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