NEB Class 12 Integration in One Shot || Mastering NEB Class 12 Integration: Complete Solution

Introduction

This tutorial aims to guide you through the concepts of integration as outlined in the NEB Class 12 curriculum. By mastering integration techniques, including anti-derivatives and partial integration methods, you will enhance your understanding of mathematics and prepare for your examinations. This step-by-step guide will break down complex topics into manageable sections, making it easier for you to learn and apply these concepts.

Step 1: Understand the Basics of Integration

• Definition: Integration is the process of finding the integral of a function, which can be thought of as the reverse operation of differentiation.

Key Concepts

Definite and Indefinite Integrals

• Definite integrals have specific limits and produce a numerical value.
• Indefinite integrals do not have limits and result in a family of functions.
• Notation: The integral of a function f(x) is denoted as ∫f(x)dx.

Step 2: Learn Anti-Derivatives

• Purpose: Anti-derivatives are used to find the original function from its derivative.

Common Anti-Derivatives

• For f(x) = x^n, the anti-derivative is (x^(n+1))/(n+1) + C, where C is the constant of integration.

Familiarize yourself with common forms like

• ∫x^n dx = (1/(n+1))x^(n+1) + C
• ∫e^x dx = e^x + C
• ∫sin(x) dx = -cos(x) + C
• ∫cos(x) dx = sin(x) + C

Step 3: Apply the Power Rule of Integration

Power Rule

To integrate functions of the form x^n, use the formula

• ∫x^n dx = (1/(n+1))x^(n+1) + C

Example

For ∫x^3 dx

• Apply the power rule: (1/(3+1))x^(3+1) + C = (1/4)x^4 + C

Step 4: Master Partial Integration

• Purpose: Partial integration is useful for integrating products of functions.

Formula

• ∫u dv = uv - ∫v du

Steps

1. Choose u and dv from the integrand.
2. Differentiate u to find du.
3. Integrate dv to find v.
4. Substitute into the formula.

Step 5: Tackle Common Integration Problems

Example Problems

Integrate ∫x * e^x dx using partial integration

• Let u = x and dv = e^x dx.
• Then du = dx and v = e^x.
• Apply the formula: ∫x * e^x dx = x * e^x - ∫e^x dx = x * e^x - e^x + C.

Conclusion

By following these steps, you will gain a solid foundation in integration, including understanding anti-derivatives and using partial integration. Practice these concepts with various functions to enhance your proficiency. As you prepare for your NEB Class 12 exams, consider working through problem sets and past papers to solidify your knowledge. Keep learning and applying these techniques to master integration!