Hey there, future mathematicians! Ready to dive into the world of integrals with Class 12 Maths Chapter 7.2? This chapter is all about integration by substitution, a super handy technique to solve tricky integrals. In this comprehensive guide, we'll break down the concepts, provide step-by-step solutions in Hindi, and make sure you ace those exams! So, grab your notebooks, and let's get started. We are going to explore all the concepts of Class 12 Maths Chapter 7.2 Solutions in Hindi in a detailed manner. This will include not only the solutions to the questions given in the textbook but also the different approaches to solve problems. Let’s get started with our exploration of Class 12 Maths Chapter 7.2 Solutions in Hindi, focusing on the concepts and methods needed to solve the problems. The chapter mainly focuses on integration by the substitution method. We will look at how to choose the right substitution, how to integrate after the substitution, and how to arrive at the final answer. The ability to solve these types of questions will equip you with the fundamental knowledge to solve the complex problems of higher classes. We also plan to look at how to approach these problems strategically, offering tips and tricks to improve your solving time and accuracy. Furthermore, we’ll see how you can apply these concepts to real-world scenarios, making the learning process both engaging and practical. Let’s not forget the importance of practicing. So, each section will contain ample exercises and practice questions. So, ready to become a master of integration? Let's get started with Class 12 Maths Chapter 7.2 Solutions in Hindi, your ultimate guide to mastering integration by substitution.

Understanding Integration by Substitution

Alright, guys, before we jump into the solutions, let's make sure we've got the basics covered. Integration by substitution (or sometimes called u-substitution) is like a clever trick to simplify integrals that look complicated at first glance. The main idea? Introduce a new variable (usually 'u') to replace a part of the integral, making it easier to integrate. Think of it as a mathematical disguise! You are going to replace a complicated part of the function with a simpler variable, like 'u'. The goal is to transform the given integral into a form that can be directly integrated using standard formulas. In this chapter, you’ll learn how to choose the right substitution, differentiate it to find du, and then rewrite the entire integral in terms of u and du. After integrating with respect to u, don't forget to substitute the original expression back in for u to get your final answer in terms of the original variable. This method is particularly useful when the integrand is a composite function or when the integral contains a function and its derivative. Master these concepts of Class 12 Maths Chapter 7.2 Solutions in Hindi, and you'll be well on your way to conquering calculus! By selecting the correct substitution, you simplify the expression and convert the given integral into a more manageable form. Let's dig deeper into the actual steps needed to solve these problems.

Steps Involved in Substitution Method:

1. Choose your u: Look for a part of the integral whose derivative is also present (or can be easily adjusted) in the integral. This could be a function inside a parenthesis, the exponent of an exponential function, or something else. This step is often the trickiest, but with practice, it'll become second nature. You need to identify a part of the integrand that, when differentiated, closely resembles another part of the integrand. The choice of 'u' is crucial; it should simplify the integral. The choice of ‘u’ is critical, and practice is the key to identifying the right substitution. Let's delve into the detailed steps.
2. Find du: Differentiate your chosen u with respect to the original variable (usually x). This will give you du/dx. Rewrite this to isolate du.
3. Rewrite the Integral: Replace the part of the integral that you chose to be u, and also replace the dx with what you found for du. Make sure the entire integral is in terms of u and du.
4. Integrate: Integrate the new integral with respect to u. This should be much easier now!
5. Substitute Back: Finally, substitute the original expression back in for u to get your answer in terms of the original variable.

Following these steps, you can simplify the integral and obtain an answer that is easily understood. Practice is the key to mastering these steps. Let’s illustrate these steps with a few examples and solve them. Remember, the goal is always to make the integral simpler.

Solutions to Exercise 7.2 (NCERT)

Alright, let's get our hands dirty with some actual problems! We'll go through the solutions to Exercise 7.2 from the NCERT textbook, step by step, in Hindi. This is where the rubber meets the road, so pay close attention. We will be going through the NCERT exercise questions step-by-step so that you have a comprehensive understanding of how to solve the problems. I encourage you to try to solve these questions yourself first, and then check with the solutions given. It is important to know the steps to find the perfect solution to the questions. We will go through each question explaining the steps to help you solve these problems. Ready, set, let’s solve some integrals! We will cover all the questions of Exercise 7.2 in the NCERT textbook, which means you will be able to solve the given questions after understanding these concepts. Following these detailed solutions will not only help you in your exams but also build a strong foundation for future mathematical studies. By going through the solutions, you will get a clear understanding of how to apply the substitution method effectively. Let's get started with Class 12 Maths Chapter 7.2 Solutions in Hindi.

Question 1:

Integrate ∫2x / (1 + x²) dx

1. Choose u: Let u = 1 + x²
2. Find du: Differentiating u with respect to x, we get du/dx = 2x, which simplifies to du = 2x dx.
3. Rewrite the Integral: The integral becomes ∫du/u.
4. Integrate: ∫du/u = ln|u| + C
5. Substitute Back: Replace u with (1 + x²): ln|1 + x²| + C. Since 1 + x² is always positive, we can write it as ln(1 + x²) + C.

Question 2:

Integrate ∫(ln x)² / x dx

1. Choose u: Let u = ln x.
2. Find du: du/dx = 1/x, which simplifies to du = (1/x) dx.
3. Rewrite the Integral: The integral becomes ∫u² du.
4. Integrate: ∫u² du = u³/3 + C.
5. Substitute Back: Replace u with ln x: (ln x)³/3 + C.

Question 3:

Integrate ∫(1 / (x + x ln x)) dx

1. Choose u: Let u = 1 + ln x.
2. Find du: du/dx = 1/x, which simplifies to du = (1/x) dx.
3. Rewrite the Integral: The integral becomes ∫du/u.
4. Integrate: ∫du/u = ln|u| + C.
5. Substitute Back: Replace u with (1 + ln x): ln|1 + ln x| + C.

Question 4:

Integrate ∫sin x sin (cos x) dx

1. Choose u: Let u = cos x.
2. Find du: du/dx = -sin x, which simplifies to -du = sin x dx.
3. Rewrite the Integral: The integral becomes -∫sin u du.
4. Integrate: -∫sin u du = cos u + C.
5. Substitute Back: Replace u with cos x: cos (cos x) + C.

Question 5:

Integrate ∫sin(ax + b) cos(ax + b) dx

1. Choose u: Let u = sin(ax + b).
2. Find du: du/dx = a cos(ax + b), which simplifies to du/a = cos(ax + b) dx.
3. Rewrite the Integral: The integral becomes ∫u/a du.
4. Integrate: ∫u/a du = u²/2a + C.
5. Substitute Back: Replace u with sin(ax + b): sin²(ax + b) / 2a + C.

Tips and Tricks for Success

Here are some handy tips and tricks to make your integration by substitution journey smoother:

• Practice, Practice, Practice: The more problems you solve, the better you'll get at recognizing the right substitutions.
• Look for Derivatives: Always keep an eye out for a function and its derivative within the integral.
• Start Simple: Begin with the basic problems and gradually move to the more complex ones.
• Double-Check: After substituting back, differentiate your answer to make sure it matches the original integrand.
• Don't Give Up: Some integrals might seem tough, but don't be discouraged. Try different substitutions or simplify the expression.
• Understand the Formulas: Remember the basic integration formulas. They are the building blocks of integration.

By following these tips, you can improve your ability to solve these problems. These tips will help you not only in exams but also in your overall understanding of the topics. Consistency and perseverance are the keys to mastery. To excel in the exam, you need to understand the concepts and practice the problems repeatedly. Regular practice and a clear understanding of the concepts are the keys to success. Always start from the basic problems and then move to the complex ones. Regular practice helps you improve your speed and accuracy. Remember, math is a skill that improves with consistent effort. So, work consistently, and do not hesitate to ask for help when you need it.

Conclusion: Mastering Chapter 7.2

So, there you have it, guys! We've journeyed through the intricacies of integration by substitution in Class 12 Maths Chapter 7.2. You've learned the fundamental concepts, seen step-by-step solutions in Hindi, and gained some valuable tips and tricks. With dedication and practice, you'll master this technique and be well-prepared for your exams and future studies. Remember to revisit the examples, solve more problems, and don't hesitate to ask for help from your teachers or peers if you get stuck. Keep practicing regularly. Keep revisiting the solutions and examples. Now go forth and conquer those integrals! With consistent practice and dedication, you can surely master the concepts of Class 12 Maths Chapter 7.2 Solutions in Hindi. Good luck with your studies, and keep up the great work! Always remember that the key to success is consistent effort and perseverance. So, all the best for your exams and future endeavors. Happy integrating!