# Integrals 12th Notes - Free NCERT Class 12 Maths Chapter 7 Notes - Download PDF

Integrals belong to the 7 chapter of NCERT. The NCERT Class 12 Maths Chapter 7 notes entirely cover up the main portions of the chapter integrals. Class 12 Maths chapter 7 notes are predominantly focused on the important formulas and their needed derivations. A Class 12 Maths chapter 7 notes enable you to find the entire chapter in an easy way. Notes for Class 12 Maths chapter 7 is made by following the sequence of the chapter starting from the indefinite integrals to definite integrals. NCERT Notes for Class 12 Maths chapter 7 not only covers the NCERT notes but covers CBSE Class 12 Maths chapter 7 notes also.

After going through Class 12 integrals notes students can also refer to,

**Also Read :**

- NCERT Solutions for Class 12 Maths Chapter 7 Integrals
- NCERT Exemplar Class 12 Maths Chapter 7 Solutions Integrals

Adding or summing up the small parts to find the whole sum of the function, is the basic meaning of integral calculus.

Indefinite integral and definite integral are the two types of integrals that are found in integral calculus.

**Indefinite Integral**

Thus the formula that gives this anti-derivative is called indefinite integral.

Integration as an Inverse Process of Differentiation

Integration is defined as the inverse of differentiation. Here we are given a function and asked to find it is primitive (the original function). Such a process is called anti-derivative or indefinite integral.

Let us see an example:-

We know that

Or

Here the actual function is “sinx +c” using indefinite integral we found the actual equation from the derived one.

Thus we came to know that the integration of some functions is written as

Here f(x) is “integrand”, f(x)dx is “element of integration” , x is “variable of integration” and “c” is the integration constant.

Some of the formulas are listed below which will help us to solve the problems.

**Geometrical Interpretation of Indefinite Integral**

Let us assume that y=f(x) be a curve such that

Now if we give values like 4, 3, 2, 1, 0, -1, -2, -3 to arbitrary constant c then the equation will give

having loci as a parabola.

Thus the equation represents a family of curves.

Some properties of indefinite integral

Here we shall derive some properties of indefinite integrals.

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

And

where c is the arbitrary constant.

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

(V) Using property (III) and (IV) can be generalised to a finite number of functions f1, f2, f3, …, fn and the real numbers k1, k2, k3, …, kn gives

Example

Find the anti-derivative (integral) of

Solution:

**Methods of Integration**

We have many ways of solving integration. Mostly we solve by using

Integration by substitution

Integration by Partial Fraction

Integration by Parts

**Integration by Substitution**

The given integral can be transferred into another form by changing the independent variable x to t by substituting x = g (t).

Let us assume that

Substitute

x=g(t) , so that dx/dt=g'(t)

Then we write dx=g’(t) dt

Example: Integrate w.r.t x, sin mx

Solution:

We know that by differentiating mx we get m. Thus we can substitute mx=t so that mdx =dt

Integration by Trigonometric Identities

The identities are used to find the integral when integration involves some trigonometric functions.

Example:

Solution: We know that

Using this we get that

**Integrals of Some Particular Function**

Some of the formulas are listed below

**Integration by Partial Fraction**

The integration of rational function can be solved by using partial fraction.

S. No | Form of a rational function | formation of partial function |

1 | ||

2 | ||

3 | ||

4 | ||

5 | ||

6 |

**Integration by Parts**

ILATE rule needs to be followed while doing integration by parts. According to ILATE rule, we can determine which function will be the 1st function and which will be the 2nd function.

ILATE

I → Inverse function

L → Logarithmic function

A → Algebraic function

T → Trigonometric function

E → Exponential function

From the above rule, as an example, we can say that there are two functions like trigonometric function and logarithmic function. The logarithmic function will be treated as the first function and the trigonometric function will be treated as the second function.

The general expression follows as

“The integral of the product of two functions = (1st function) × (integral of the 2nd function) – Integral of [(differential coefficient of the 1st function) × (integral of the 2nd function)]”

Example:

Integration of special types function

The formulas are listed below

**Definite Integral**

In indefinite, we saw that the result of the integration was not a unique value. But here indefinite integral we will find a unique value and it is well defined in a limited boundary. Generally, definite integral is denoted by

Her “a” is the lower limit and “b” is the upper limit of the integral. The value of the definite integral is computed by F(b) - F(a).

Basically in Definite integral consist of two cases as discussed below:

**Definite Integral as a Limit of the Sum**

If the definite integral

is the area bounded by the curve y = f(x), the ordinates x = a, x = b and the x-axis. To evaluate this area, consider the region PRSQP between this curve, x-axis, and the ordinates x = a and x =b

From the figure, we find the general equation of definite integral as a limit of sum

**Some Properties of Definite Integral**

Let us see an example

This brings us to the end of the chapter.

**Significance of NCERT Class 12 Maths Chapter 7 Notes**

Class 12 integrals notes will be really helpful to revise the chapter and get a brief overview of the important topics. Also, Class 12 Math Chapter 7 Notes is useful for covering Class 12 CBSE syllabuses and also for competitive exams like BITSAT, JEE MAINS. Class 12 Math chapter 7 notes pdf download can be used for preparing in offline mode.

integrals Class 12 notes pdf download: link

**NCERT Class 12 Notes Chapter Wise.**

**Subject Wise NCERT Exemplar Solutions**

- NCERT Exemplar Class 12 Solutions
- NCERT Exemplar Class 12 Maths
- NCERT Exemplar Class 12 Physics
- NCERT Exemplar Class 12 Chemistry
- NCERT Exemplar Class 12 Biology

**Subject Wise NCERT Solutions**

- NCERT Solutions for Class 12 Mathematics
- NCERT Solutions for Class 12 Chemistry
- NCERT Solutions for Class 12 Physics
- NCERT Solutions for Class 12 Biology

## Frequently Asked Question (FAQs) - Integrals 12th Notes - Free NCERT Class 12 Maths Chapter 7 Notes - Download PDF

**Question: **From where we can download this notes>

**Answer: **

The notes can be downloaded from integrals Class 12 notes pdf download.

**Question: **What textbook should be followed?

**Answer: **

Class 12 math chapter 7 textbook should be followed by Class 12 Maths chapter 7 notes pdf download

**Question: **Write the different types of integrals in Maths.

**Answer: **

There are two types of Integration. They are definite integral and indefinite integral.

**Question: **Define the fundamental theorem of calculus?

**Answer: **

The fundamental theorem of calculus connects the concept of differentiation and integration of a function.

**Question: **What are the uses of integration?

**Answer: **

In, NCERT Class 12 Maths chapter 7, integration is used to find the area, volumes, and central values of many things.

**Question: **Define integration?

**Answer: **

The integration is the process of finding the anti-derivative function. Integration is defined as the reverse process of differentiation.

## Latest Articles

##### TOSS Result 2023 for SSC & Inter - Telangana Open School Socie...

TOSS Result 2023 - Telangana Open School Society releases the ...

##### Maharashtra HSC Result 2023 - Check Maharashtra board 12th res...

Maharashtra HSC Result 2023 - Maharashtra Board will announce ...

##### CBSE 12th Result 2023 - CBSE Class 12th Result at cbseresults....

CBSE Class 12 Result 2023 - Central Board of Secondary Educati...

##### IEO Result 2022-23 (Released)- Check SOF IEO result at sofworl...

IEO Result 2022-23 - Know where to check SOF IEO result 2022-2...

##### NSO Result 2022-23 - Check SOF NSO Result for Level 1 & 2 Here

NSO Result 2022-23 - Science Olympiad Foundation has released ...

##### IMO Results 2022-23 (Released) - Check SOF IMO Level 1 & 2 Res...

IMO Results 2022-23 - Science Olympiad Foundation has released...

##### TN 12th Result 2023 - Check Tamil Nadu Class 12 Result for Sci...

TN 12th Result 2023 - Directorate of Government Examinations r...

##### TN 11th Result 2023 - Plus One Tamil Nadu Result @tnresults.ni...

TN 11th Result 2023 Tamilnadu - DGE will release the Tamil Nad...

##### CGBSE 12th Result 2023 - Chhattisgarh Board Class 12th Result ...

CGBSE 12th Result 2023 - Chhattisgarh Board will release the C...

##### UK Board 12th Result 2023 - Uttarakhand Board Class 12th Resul...

UK Board 12th Result 2023- Uttarakhand Board of School Educati...