NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.10 Integrals - PDF Download

1. Some Properties of Definite Integrals

A few significant properties of definite integrals are listed below. These will make it easier to evaluate the definite integrals.

$\mathbf{P}_0: \quad \int_a^b f(x) d x=\int_a^b f(t) d t$

$\mathbf{P}_1: \quad \int_a^b f(x) d x=-\int_b^a f(x) d x$ . In particular, $\int_a^a f(x) d x=0$

$\mathbf{P}_2: \quad \int_a^b f(x) d x=\int_a^c f(x) d x+\int_c^b f(x) d x$

$\mathbf{P}_3: \quad \int_a^b f(x) d x=\int_a^b f(a+b-x) d x$

$\mathbf{P}_4: \quad \int_0^a f(x) d x=\int_0^a f(a-x) d x$

(Note that P4 is a particular case of P3 )

$\mathbf{P}_5: \quad \int_0^{2 a} f(x) d x=\int_0^a f(x) d x+\int_0^a f(2 a-x) d x$

$\mathbf{P}_6: \quad \int_0^{2 a} f(x) d x=2 \int_0^a f(x) d x$, if $f(2 a-x)=f(x)$ and 0 if $f(2 a-x)=-f(x)$

$\mathbf{P}_7:$ (i) $\int_{-a}^a f(x) d x=2 \int_0^a f(x) d x$ , if f is an even function, i.e., if f (– x) = f (x).

                         (ii) $\int_{-a}^a f(x) d x=0$ , if f is an odd function, i.e., if f (– x) = – f (x).

You can check the proofs for each of these properties in NCERT textbook class 12 maths chapter 7 ex 7.10. 

Tips for Solving Exercise 7.10 Class 12 Chapter 7 Integrals

NCERT solutions class 12 maths chapter 7 exercise 7.10 will provide you the understanding for all the properties of definite integrals to find out the simplest way to integrate them. eSaral’s expert faculty have come up with some important tips for solving ex 7.10 class 12 maths.

1. You need to remember to do a thorough analysis of each problem before attempting it in order to prevent calculation errors.

2. Students will be able to clear up all of their questions and perform well in exams by understanding all of the definitions and proofs of the formulas included in NCERT solutions class 12 maths chapter 7 ex 7.10.

3. By studying these NCERT solutions, students will get well-versed with the pattern of numerous problems asked in exams.

Importance of Solving Ex 7.10 Class 12 Maths Chapter 7 Integrals

There are numerous advantages of solving ex 7.10 class 12 maths chapter 7.

1. NCERT solutions for ex 7.10 class 12 maths ch 7 has questions based on properties of definite integrals. These properties are explained sequencely by eSaral's expert teachers that will help you prepare for exams.

2. In NCERT solutions class 12 maths chapter 7 ex 7.10 includes examples that will give you proper understanding of concepts.

3. By practicing questions in NCERT solutions for ex 7.10 class 12 maths will help you comprehend the properties more effectively.

4. NCERT solution PDFs will give you an opportunity to cross check your answer that will boost your confidence.  
   

Frequently Asked Questions

Question 1. Write 7th and 8th properties of definite integrals?

Answer 1. The properties 7th and 8th of definite integrals are as follows.

Property 7: $\int_0^{2 a} f(x) d x=2 \int_0^a f(x) d x$, if $f(2 a-x)=f(x)$ and 0 if $f(2 a-x)=-f(x)$

Property 8: (i) $\int_{-a}^a f(x) d x=2 \int_0^a f(x) d x$ , if f is an even function, i.e., if f (– x) = f (x).

                  (ii) $\int_{-a}^a f(x) d x=0$ , if f is an odd function, i.e., if f (– x) = – f (x). 

Question 2. Can I download the NCERT solution PDFs for ex 7.10 class 12 maths chapter 7?

Answer 2. Yes, you can download the NCERT solution PDFs for ex 7.10 class 12 maths chapter 7 from the official website of eSaral.