NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.1 Integrals - PDF Download

NCERT solutions for class 12 maths chapter 7 exercise 7.1 Integrals requires us to determine the integration as an inverse process of differentiation and some properties of indefinite integral. Since differentiation and integration are closely connected, NCERT solutions developed by subject experts of eSaral help students understand the differences between the two processes so they can better understand the concepts. Solutions of ex 7.1 class 12 maths chapter 7 are created in simple language to give you better understanding of topics and concepts.

There are a total of 22 questions in exercise 7.1 class 12 maths ch 7 in which some of the questions are simple to solve and some questions require a better comprehension of properties of indefinite integral. NCERT solutions are provided here for you to comprehend the basic concepts of topics mentioned in ex 7.1 class 12 maths that help in preparing for exams. You can also download the Pdf version of these solutions to practice questions. The link to download the PDF for free is given below.  

Topics Covered in Exercise 7.1 Class 12 Mathematics Questions

Ex 7.1 class 12 maths solutions covers the topics of integration as an inverse process of differentiation and some properties of indefinite integral which are explained below.

1. Integration as an Inverse Process of Differentiation

Integration is the inverse process of differentiation. Instead of differentiating a function, we are given the derivative of a function and asked to find its primitive, i.e., the original function. Such a process is called integration or anti differentiation.

Let’s take a example $\frac{d}{d x}(\sin x)=\cos x$  …..(1)

                       $\frac{d}{d x}\left(\frac{x^3}{3}\right)=x^2$   …..(2)

And               $\frac{d}{d x}\left(e^x\right)=e^x$   ….(3)

The function cos x in (1) is the derived function of sin x, as we can see. Sin x is referred to as the antiderivative (or integral) of cos x. Similar to this, in (2) and (3), x33 and ex, respectively, represent the anti derivatives (or integrals) of x2 and ex. 

Again, we note that for any real number C, treated as constant function, its derivative is zero and hence, we can write (1), (2) and (3) as follows :

$\frac{d}{d x}(\sin x+\mathrm{C})=\cos x, \frac{d}{d x}\left(\frac{x^3}{3}+\mathrm{C}\right)=x^2$ and $\frac{d}{d x}\left(e^x+\mathrm{C}\right)=e^x$

Consequently, anti derivatives (or integrals) of the functions mentioned above are not unique. Actually, there exist infinitely many anti derivatives of each of these functions which can be obtained by choosing C arbitrarily from the set of real numbers. For this reason C is customarily referred to as an arbitrary constant. In fact, C is the parameter by varying which one gets different antiderivatives (or integrals) of the given function.

More generally, if there is a function F such that $\frac{d}{d x} \mathrm{~F}(x)=f(x), \forall x \in \mathrm{I}$ (interval), then for any arbitrary real number C, (also called constant of integration)

$\frac{d}{d x}[\mathrm{~F}(x)+\mathrm{C}]=f(x), x \in \mathrm{I}$

Thus, {F + C, C ∈ R} denotes a family of antiderivatives of f.

Some properties of indefinite integral

Some properties of indefinite integrals will be derived in this subsection.

(I) The process of differentiation and integration are inverses of each other in the sense of the following results :

       $\frac{d}{d x} \int f(x) d x=f(x)$ and 

$\int f^{\prime}(x) d x=f(x)+\mathrm{C}$ , where C is any arbitrary constant.

(II) Two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent.

(III) $\int[f(x)+g(x)] d x=\int f(x) d x+\int g(x) d x$

(IV) For any real number k, $\int k f(x) d x=k \int f(x) d x$

(V) Properties (III) and (IV) can be generalised to a finite number of functions f1 , f2 , ..., fn and the real numbers, k1 , k2 , ..., kn giving

$\begin{aligned} & \int\left[k_1 f_1(x)+k_2 f_2(x)+\ldots+k_n f_n(x)\right] d x \\ & =k_1 \int f_1(x) d x+k_2 \int f_2(x) d x+\ldots+k_n \int f_n(x) d x .\end{aligned}$

Tips for Solving Exercise 7.1 Class 12 Chapter 7 Integrals

Ex 7.1 class 12 maths solutions combined useful and important tips by experts of eSaral to solve questions with ease.

1. The questions require more observation than calculation. However, here we provide lists of several formulas that students must understand and remember.

2. There are two main properties in ex 7.1 class 12 maths chapter 7. Students should make it a point to revise the proofs for both of them in order to gain a deeper comprehension of the topic.

3. Students should also solve the examples given in ex 7.1 class 12 maths to be able to understand the concepts and theorems properly. 

Importance of Solving Ex 7.1 Class 12 Maths Chapter 7 Integrals 

You will get a lot of benefits by solving ex 7.1 class 12 maths  chapter 7. There are some of the benefits provided by expert teachers of eSaral which you can check below.

1. NCERT solutions class 12 maths chapter 7 ex 7.1 is about integration as an inverse process of differentiation. These topics have been explained by subject experts of eSaral to help you solve the ex 7.1 without any confusion. 

2. Ex 7.1 class 12 maths solutions have provided precise knowledge of properties and theorems that is essential for understanding the nature of questions.

3. These solutions are also available in PDF format that will help you grasp the accurate answers to all the questions of exercise 7.1.

4. Students must practice the exercise and revise the concepts for better comprehension of questions asked in board exams.

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