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

JEE Mains & Advanced

NCERT solutions for class 12 maths chapter 7 exercise 7.10 Integrals describes some properties of definite integrals that make it easier to evaluate the definite integrals. To better comprehend this exercise, you should go through the methods used to prove these properties. The academic team of mathematics at eSaral have provided thorough knowledge of these properties to solve the questions of ex 7.10 class 12 maths chapter 7.

Class 12 maths chapter 7 exercise 7.10 NCERT solutions has a total of 21 questions and the difficulty level of these questions are medium to difficult. To solve these problems, you must simplify the function using properties explained by subject experts of eSaral and transform it into a format that is simpler to integrate. You should also memorize these properties as formulas to solve questions in exams quickly. Ex 7.10 class 12 maths solutions are also provided in PDF format. You can download these free PDFs on the eSaral website and practice all the questions to score high marks in exams. Download the NCERT solution PDF for ex 7.10 class 12 maths ch 7 from the link given below.

## Topics Covered in Exercise 7.10 Class 12 Mathematics Questions

Ex 7.10 class 12 maths chapter 7 is based on some properties of definite integrals. You can check the properties which are mentioned below.

 1 Some Properties of Definite Integrals
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.

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