Hey guys! Are you struggling with Class 12 Maths, especially Chapter 7? No worries, you're not alone! This chapter can be a bit tricky, but with the right approach and resources, you can totally nail it. In this guide, we'll break down the solutions in Hindi to help you understand each concept clearly. Let's dive in!

Understanding Chapter 7

Before we jump into the solutions, let's quickly recap what Chapter 7 is all about. Chapter 7 typically covers integrals, which is a fundamental concept in calculus. Integrals are essentially the reverse process of differentiation. You'll learn about indefinite and definite integrals, various methods of integration, and their applications. Understanding the basic formulas and techniques is crucial for solving the exercises.

Key Concepts in Integrals

• Indefinite Integrals: These are integrals without limits, resulting in a general function plus a constant of integration (C). The constant C represents the family of functions whose derivative is the integrand.
• Definite Integrals: These integrals have upper and lower limits, providing a specific value representing the area under the curve between those limits. The Fundamental Theorem of Calculus connects definite and indefinite integrals.
• Methods of Integration:
  • Substitution: Simplifying the integral by substituting a part of the integrand with a new variable.
  • Integration by Parts: Used for integrating products of functions, using the formula ∫u dv = uv - ∫v du.
  • Partial Fractions: Decomposing rational functions into simpler fractions for easier integration. This method is especially useful when dealing with rational functions where the degree of the numerator is less than the degree of the denominator.
• Applications of Integrals: Finding areas under curves, volumes of solids, average values, and more. Integrals are not just theoretical concepts but have practical applications in physics, engineering, economics, and other fields.

Mastering these concepts will not only help you solve the exercises but also build a strong foundation for future studies in mathematics and related fields. Make sure to practice regularly and seek help when needed to fully grasp the concepts.

Exercise Solutions in Hindi

Now, let's get to the heart of the matter: solving the exercises. We'll focus on providing solutions in Hindi so that you can understand each step clearly. Remember, the goal is not just to memorize the steps, but to understand the logic behind them. By understanding the logic, you'll be able to apply these techniques to a wide range of problems.

Example 1: Indefinite Integral

Question: Evaluate ∫x^2 dx.

Solution:

हम जानते हैं कि ∫x^n dx = (x^(n+1))/(n+1) + C.

इसलिए, ∫x^2 dx = (x^(2+1))/(2+1) + C = (x^3)/3 + C.

This solution uses the power rule for integration. The power rule states that the integral of x raised to the power of n is x raised to the power of n+1, divided by n+1, plus the constant of integration C. Understanding and memorizing these basic rules is essential for tackling more complex integrals.

Example 2: Definite Integral

Question: Evaluate ∫[0,1] x dx.

Solution:

पहले, हम अनिश्चितकालीन अभिन्न ∫x dx ढूंढते हैं, जो (x^2)/2 + C है।

फिर, हम सीमाओं का उपयोग करते हैं: [(1)^2/2] - [(0)^2/2] = 1/2 - 0 = 1/2.

This example demonstrates how to evaluate a definite integral. First, you find the indefinite integral, and then you evaluate it at the upper and lower limits of integration. The difference between these values gives you the definite integral.

Example 3: Integration by Substitution

Question: Evaluate ∫sin(x)cos(x) dx.

Solution:

मान लीजिए u = sin(x), तो du = cos(x) dx.

इसलिए, ∫sin(x)cos(x) dx = ∫u du = (u^2)/2 + C = (sin^2(x))/2 + C.

In this example, we use the substitution method. By substituting sin(x) with u, we simplify the integral and make it easier to solve. Remember, the key to using the substitution method is to choose a suitable substitution that simplifies the integrand.

Example 4: Integration by Parts

Question: Evaluate ∫x sin(x) dx.

Solution:

मान लीजिए u = x, dv = sin(x) dx.

तो, du = dx, v = -cos(x).

सूत्र ∫u dv = uv - ∫v du का उपयोग करते हुए,

∫x sin(x) dx = -x cos(x) - ∫-cos(x) dx = -x cos(x) + ∫cos(x) dx = -x cos(x) + sin(x) + C.

Integration by parts is useful for integrating products of functions. The key is to choose u and dv wisely. In this case, choosing u = x and dv = sin(x) dx simplifies the integral.

Tips for Solving Integrals

• Know Your Formulas: Memorize basic integration formulas. The more formulas you have at your fingertips, the easier it will be to solve problems.
• Practice Regularly: The more you practice, the better you'll become at recognizing patterns and choosing the right methods.
• Break Down Complex Problems: Break complex integrals into simpler parts. Use algebraic manipulation, trigonometric identities, and other techniques to simplify the integrand.
• Check Your Answers: Differentiate your result to see if you get back the original integrand. This is a good way to check your work and ensure that you haven't made any mistakes.

Resources for Further Help

If you're still struggling, don't worry! There are plenty of resources available to help you out.

Textbooks and Notes

Refer to your textbook and class notes. Pay close attention to the examples and explanations provided by your teacher.

Online Tutorials

Websites like Khan Academy, YouTube, and Vedantu offer excellent tutorials and explanations on integration. These resources often provide visual aids and step-by-step solutions that can be very helpful.

Study Groups

Join a study group with your classmates. Working together can help you understand the material better and learn from each other.

Teacher Assistance

Don't hesitate to ask your teacher for help. They can provide personalized guidance and answer any questions you may have.

Conclusion

So, there you have it! Solving Class 12 Maths Chapter 7 in Hindi doesn't have to be a daunting task. With a solid understanding of the concepts, regular practice, and the right resources, you can conquer integrals and ace your exams. Keep practicing, stay focused, and remember to take breaks when you need them. You've got this!

If you found this guide helpful, be sure to share it with your friends and classmates. Good luck with your studies, and keep rocking those maths problems!