Introduction to NCERT Solutions for Class 12 Maths Chapter 7
NCERT Solutions for Integrals class 12 Maths will provide the proper knowledge of the integrals. This chapter is all about the computation of integrals. These NCERT Solutions gives you a strong calculative skill for finding the integrals values of different functions. It will also help you with your homework and best performance in exams.
NCERT Solutions for Class 12 Maths Chapter 7-Integrals
NCERT solutions will help the students to understand the purpose of definite integrals by applying it on real problems. It presents the solutions in a very effective and systematic way. The student learns many elementary methods to do the integration. The conceptual background of integrals is very much necessary for other branches of mathematics as well as science.This chapter contains many examples and problems to give maximum practice to the students.Let us discuss the sub-topics in detail.This chapter will introduce the indefinite integral of the function, which is the anti-derivative process. It is also known as integration.
Integration as an Inverse Process of Differentiation
It is the reverse process of differentiating a function because we will have a derivative of a function and then we will find its primitive, i.e., the original function.
- Geometrical interpretation of indefinite integralThis topic explains about the geometrical representation of integration i.e. graph for integrals.
- Some properties of indefinite integral:This topic is about some interesting properties of indefinite integrals and also about integrals of some popular functions.
- Comparison between differentiation and integration:The student will see the comparison between differentiation and integration based on various properties.
Methods of Integration
Previous method to find integrals are not suitable always. Hence, in this topic, we need to develop additional methods for finding the integrals with a reduction to standard forms.
- Integration by substitution:In this section, the student will learn the method of integration by substitution in an easy way.
- Integration using trigonometric identities:When the integrand involves some trigonometric functions, then the student can use some known identities to find the integral.
Integrals of Some Particular Functions
In this topic, the student will learn some important formulae of integrals and they will apply them for integrating many other related standard integrals.
It is always possible to write the integrand as the addition of simpler rational functions by a method called partial fraction decomposition. Then the process of integration becomes very easy and usual.
Integration by Parts
The student will learn an important method of integration, which very useful in integrating products of two functions.
- Integral of the type ∫ [f( x ) f ’( x ) ] dx :This section will explain the method to find the value of integration of a specific type.
- Integrals of some more types:It discusses some special types of standard integrals based on the technique of integration by parts.
In this section, the student will study a definite integral of a function. The definite integral has a unique value. The definite integral works as the limit of a sum.
- Definite integral as the limit of a sum:This is the fundamental method to find definite integrals i.e. by using limit of sum method.
Fundamental Theorem of Calculus
This section is about fundamental theorem for finding the integration.
- Area function:This states the integration is the area of the region bounded by the curve y = f (x), the coordinates x = a, x = b, and x-axis.
- First fundamental theorem of the integral calculus:It explains First fundamental theorem of integral calculus.
- Second fundamental theorem of the integral calculus:It explains the Second fundamental theorem of integral calculus.
Evaluation of Definite Integrals by Substitution
One of the important methods for finding the definite integral is the method of substitution, similar for finding indefinite integrals.
Some Properties of Definite Integrals
Definite integrals are very important and used in maths as well as in science for various applications. This section discusses some important methods and formula for finding definite integrals.