NCERT Solutions for Class 12 Maths Chapter 7: Integrals
The chapter NCERT Solutions for Class 12 Maths Chapter 7: Integrals consists of the concept and method involving integral calculus. This class 12 maths chapter 7 includes indefinite and definite integrals along with various integration methods like integration by substitution, integration by parts, and integration using partial fractions. The solutions have elaborated each method in detail, which shall be very helpful for the students in understanding and applying these to different problems. These solutions will help the pupils in mastering the basic course of Grade 12 integrals, but which matters further on to solve complex mathematical problems in higher studies and different competitive examinations. These solutions also include all the applications of the integrals on finding the area under curves and between two curves that is included in the NCERT textbook. The stepwise approach given in class 12 maths chapter 7 pdf would be very helpful in achieving the self-same goal that students can follow in the steps and in a position to interpret the solution themselves. These solutions form a necessary building block for a deep understanding and are strictly according to the latest CBSE syllabus. The student, if practiced, can solve the board exams and competitive entrance tests which include the concept of integrals.
Download PDF For NCERT Solutions for Maths Integrals
The NCERT Solutions for Class 12 Maths Chapter 7: Integrals are tailored to help the students master the concepts that are key to success in their classrooms. The solutions given in the PDF are developed by experts and correlate with the CBSE syllabus of 2023-2024. These solutions provide thorough explanations with a step-by-step approach to solving problems. Students can easily get a hold of the subject and learn the basics with a deeper understanding. Additionally, they can practice better, be confident, and perform well in their examinations with the support of this PDF.
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Access Answers to NCERT Solutions for Class 12 Maths Chapter 7: Integrals
Students can access the NCERT Solutions for Class 12 Maths Chapter 7: Integrals. Curated by experts according to the CBSE syllabus for 2023–2024, these step-by-step solutions make Maths much easier to understand and learn for the students. These solutions can be used in practice by students to attain skills in solving problems, reinforce important learning objectives, and be well-prepared for tests.
Integrals - Exercise 7.1
Evaluate the following integrals
∫(4e3x+ 1) dx
Find an antiderivative (or integral) of the following functions by the method of inspection in Exercises 1 to 5.
sin 2x
The anti derivative of sin 2x is a function of x whose derivative is sin 2x.
It is known that,
herefore, the anti derivative of sin 2x is -1/2 cos 2x.
cos 3x
The anti derivative of cos 3x is a function of x whose derivative is cos 3x.
It is known that,
Therefore, the anti derivative of cos 3x is 1/3 sin 3x.
Find an antiderivative (or integral) of the following functions by the method of inspection
e2x
The anti derivative of cos 3x is a function of x whose derivative is cos 3x.
It is known that,
Therefore, the anti derivative of cos 3x is 1/3 sin 3x.
e2x
The anti derivative of e2x is the function of x whose derivative is e2x.
It is known that,
Therefore, the anti derivative of e2x is 1/2 e2x.
(ax + b)2
The anti derivative of (ax + b)2 is the function of x whose derivative is (ax + b)2.
It is known that,
sin 2x – 4 e3x
The anti derivative of sin 2x – 4 e3x is the function of x whose derivative is sin 2x – 4 e3x
It is known that,
Evaluate the following integrals
Evaluate the following integrals
Evaluate the following integrals
∫(1 – x)√x dx
∫(1 – x)√x dx
Evaluate the following integrals
Choose the correct answer:
The anti derivative of 
Therefore, option (C) is correct.
Choose the correct answer:
If 
It is given that,
Therefore, option (A) is correct.
Integrals - Exercise 7.2
sec2 (7 – 4x)
Let 7 − 4x = t
∴ −4dx = dt
x√1 + 2x2
Let 1 + 2x2 = t
∴ 4xdx = dt
Integrate the functions
Integrate the functions
Let 1 + x2 = t
∴2x dx = dt
Let log |x| = t
∴ 1/x dx = dt
sin x ⋅ sin (cos x)
sin x ⋅ sin (cos x)
Let cos x = t
∴ −sin x dx = dt
sin(ax + b) cos(ax + b)
√ax + b
Let ax + b = t
⇒ adx = dt
x√x + 2
Let (x + 2) = t
∴ dx = dt
Let x3 – 1 = t
∴ 3x2 dx = dt
Let 2 + 3x3 = t
∴ 9x2 dx = dt
Let log x = t
∴ 1/x dx = dt
x/9 – 4x2
Let 9 – 4x2 = t
∴ −8x dx = dt
Let 2x + 3 = t
∴ 2dx = dt
tan2 (2x – 3)
Let x2 = t
∴ 2xdx = dt
Integrate the functions
Dividing numerator and denominator by ex, we obtain
Let √x = t
Integrate the functions
Let sin 2x = t
Let 1 + sin x = t
∴ cos x dx = dt
cot x log sin x
Let log sin x = t
sin x/1 + cos x
Let 1 + cos x = t
∴ −sin x dx = dt
sin x/(1 + cos x)2
Let 1 + cos x = t
∴ −sin x dx = dt
1/1 + cot x
1/1 – tan x
Let 1 + log x = t
∴ 1/x dx = dt
Let x4 = t
∴ 4x3 dx = dt
(A) 10x – x10 + C
(B) 10x + x10 + C
(C) (10x – x10)-1 + C
(D) log(10x + x10) + C
Therefore, option (D) is correct.
(A) tan x + cot x + C
(B) tan x – cot x + C
(C) tan x cot x + C
(D) tan x – cot 2x + C
Therefore, option (B) is correct.
Integrals - Exercise 7.3
sin 3x cos4x
sin2(2x + 5)
cos 2x cos 4x cos 6x
sin 4x sin 8x
It is known that,
sin A . sin B = 12cosA-B-cosA+B
∴∫sin4x sin8x dx=∫12cos4x-8x-cos4x+8xdx
=12∫cos-4x-cos12xdx
=12∫cos4x-cos12xdx
=12sin4x4-sin12x12+C
sin3 (2x + 1)
sin3 x cos3 x
sin x sin 2x sin 3x
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