Under Integrals Class 12 NCERT Solutions, students learn the different ways of finding the integral of a function, whose derivative is given. Therefore, integration is also called anti-differentiation. By differentiating an arbitrary constant, different integrals of any given function can be obtained. This is also called the constant of integration.

Hence, for any arbitrary real number C, ∫f(x) dx = F(x) + C.

The above equation represents an indefinite integral. In this chapter, students learn a number of essential formulas and characteristics of the indefinite integral as well as how to interpret it geometrically.

There are three different ways of integration:

• By Substitutions:
• This method is used when a variable is substituted by another appropriate and relevant variable to make the process of integration easier.
• Example: If I = ∫f(x)dx and if x = g(t), then, I = ∫ fg(t)g'(t) dt.
• If the integral of a function of x is known, then if x is multiplied by a constant and another constant is added to the product, the integral is of the same form but is divided by the coefficient of x.
• The following are some useful suggestions while using the substitution method:
• If the integrand contains a t-ratio of f(x) or logarithm of f(x) or an exponential function in which the index is f(x), put f(x) = t.
• If the integrand is the rational function of ex, put ex = t.  
• Using Partial Functions:
• Let us consider two polynomials p1(x) and p2(x) on x. Here, p2(x) <>0 and the degrees of p1(x) > p2(x).
• Then, p1(x)/ p2(x) = t(x) + p’(x)/ p2(x).
• In the above equation, t(x) is a polynomial in x which is integrated and degree of p’(x) < degree of p2(x).
• By Parts:
• Consider two functions f1(x) and f2(x).
• Then, ∫[ f1(x) f2(x)] dx = f1(x) ∫f2(x)] dx – ∫{f'(x) ∫f2(x)] dx} dx.
• Hence, the above method is not suitable for the product of functions.

In this part of the Fundamental Theorem of Integral Calculus, students learn essential integral formulas and ways to apply them for many other related standard integrals.

• ∫ dx/ (x2 – b2) = 1/2b log |x – b/x + a| + C
• ∫ dx/ (x2 + b2) = 1/b tan-1 x/b + C
• ∫ dx/ (√x2 + b2) = log |x + √x2 + b2| + C

The following are some standard integrals:

• ∫ tan x dx = -log I cos x I + C or log I sec x I + C
• ∫ cot x dx = log I sin x I + C
• ∫ sec x dx = log I sec x + tan x I + C
• ∫ cosec x dx = log I cosec x - cot x I + C or log I tan x/2 I + C

This part of Chapter 7 Integrals Class 12 further expands on integration by partial functions. Students learn to convert an improper rational function into a proper one through long division.

Improper rational function: It is a function where two polynomials p1(x) and p2(x) on x, p2(x) <>0 and degrees of p1(x) > p2(x) t.

This is also known as partial integration and teaches the methods to integrate products of functions.

Consider a variable x. For this variable, there are two differentiable functions f1 and f2.

By using the product rule of differentiation we can write d (f1f2)/dx = f1df2/dx + f2 df1/dx.

Now, by applying integration to both sides, we can deduce that the integral of the product of two functions is given as follows:

(1st function) x (integral of the 2nd function) – Integral of (differential coefficient of 1st function) x (integral of the 2nd function)

In this section, students learn about the meaning and usage of a definite integral and to denote a definite integral. Its notation is given as follows:

∫baf(x)dx

In the above equation, a is the lower limit of the denotation, and b is the upper limit of the integral.

Students also learn the way to represent a definite integral either as a limit of a sum or an integral in the interval a,b.

In the Fundamental Theorem of Integral Calculus, students learn the link between the concepts of integration and differentiation of a function. This is done by calculating the integral difference of the integration process at the upper and lower limits.

The fundamental theorems of calculus are given below:

• Area function:
• ∫ab f(x) dx defines the area of a curve y = f(x) bounded by ordinates of the x line (a and b).
• Hence, the area A(x) =∫ba f(x) dx.
• The first fundamental theorem of integral calculus: A’(x) = f(x) for all x ε a,b, where f is a continuous function on the closed interval a,b.
• The second fundamental theorem of integral calculus:
• ∫ab f(x) dx = F(x) = F(x)ba = F(b) – F(a).
• Here, f - is a continuous function on the closed interval a,b.
• F – antiderivative of f.

The steps to evaluate definite integrals ∫ab f(x) dx by substitution are as follows:

• First, we reduce the given integral to a known form by replacing y = f(x) and x = q(y). This is done by considering the integral without limits.
• Second, we integrate the new integrand without stating the constant of integration.

The following are some properties of definite integrals:

• ∫ab f(x) dx = ∫ab f(t) dt
• ∫02p f(x) dx = ∫0b f(x) dx + ∫0p f(2p - x) dx
• ∫0p f(x) dx = ∫0p f(p - x) dx
• ∫ab f(x) dx = ∫ap f(x) + ∫pb f(x) dx

• NCERT Solutions for Class 7 Maths Chapter 7 uses easy language and eye-catching formats.
• The topics are as per the latest syllabus so that the students can easily revise the notes in a minimum time with maximum accuracy.
• Integrals NCERT Solutions Class 12 integrals notes are as per the CBSE’s syllabus guidelines.
• After studying integrals NCERT Solutions Class 12 notes, students will be able to attempt questions in the books confidently.
• The Class 12 Maths NCERT Solutions Chapter 7 notes cover all necessary formulas and concepts presented in the chapter.
• These notes will evidently save the student’s time during exam preparation.

1. What is the meaning of Integrals in simple terms?

In Calculus, the area under a curve is known as an integral. When mathematical functions are represented graphically on a graph paper, the area enclosed between the curve of a function and the x-axis is the integral value of that particular equation.

Students learn to apply and work on a list of formulas to calculate various functions’ integration in Class 12 Maths.

In an integral, also known as an anti-derivative, the integration formulas for some functions are mostly the reverse of their differentiation formulas.

This is also one of the most important topics in Class 12 Mathematics.

2. What is the difference between definite and indefinite integrals?

A definite integral is one where the upper and lower limits are given for the integral. It's important to remember that while computing the definite integration of a given function f(x) for upper and lower limits a and b, you're calculating the area under the curve f(x) from x=a to x=b. Indefinite integrals do not have upper and lower bounds, unlike definite integrals. So, you can figure out a generic integral value for a bunch of similar functions f(x), in indefinite integral, where ‘x’ can have various solutions. Moreover, the notation for additive constant C is written with the indefinite integral value of any function.

3. Is it difficult to understand Class 12 Integrals?

No, integral calculus in Class 12 is not difficult to grasp; in fact, it is one of the most fascinating topics in the 12th-grade Maths curriculum. If you have a good grasp of the integration formulas for different functions, you will be able to solve the fundamental sums of integration. Also, having a good comprehension of derivatives makes it easier to understand the ideas of integral calculus. Integration is also known as the antiderivative. Therefore, it is critical to have a thorough knowledge of all integral calculus principles presented in Class 12. These ideas serve as the foundation for complex mathematical and statistical theories.

4. Where can I find important study materials related to Class 12 Maths Integrals?

You can access high-quality NCERT Solutions for Class 12 Maths Chapter 7 Integrals on MSVgo. MSVGO’s NCERT solutions are compiled by our subject-matter experts and are highly rated integral solutions among the ones available online. All of the sums of NCERT Class 12 Maths Chapter 7 are presented stepwise so that while referring to them, the students can check their own solutions and also spot their own mistakes. Our subject-matter experts have used the latest CBSE guidelines for Class 12 to put together the NCERT Maths solutions for integrals. These NCERT solutions can be downloaded for free and refer to them while practising.