## NCERT Solutions for Class 12 Maths Chapter 7 Integrals

**The topics and sub-topics included in the Integrals chapter are the following:**

Section Name | Topic Name |

7 | Integrals |

7.1 | Introduction |

7.2 | Integration as an Inverse Process of Differentiation |

7.3 | Methods of Integration |

7.4 | Integrals of some Particular Functions |

7.5 | Integration by Partial Fractions |

7.6 | Integration by Parts |

7.7 | Definite Integral |

7.8 | Fundamental Theorem of Calculus |

7.9 | Evaluation of Definite Integrals by Substitution |

7.10 | Some Properties of Definite Integrals |

**NCERT Solutions for Class 12 Maths Chapter 7 Integrals: **Engineering aspirants and students appearing for CBSE Class 12 board exams must take the NCERT Mathematics textbooks seriously and finish them from top to bottom. They must also go through the NCERT solutions for Class 12 Maths and Class 11 Maths to have a better understanding of the various concepts. In this article, we will provide you with NCERT Solutions for Class 12 Maths Chapter 7 – Integrals which have been designed by the best teachers in India.

Free download NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.11, Ex 7.10, Ex 7.9, Ex 7.8, Ex 7.7, Ex 7.6, Ex 7.5, Ex 7.4, Ex 7.3, Ex 7.2, Ex 7.1 Integrals PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

- Integration Formulas
- Integrals Class 12 Ex 7.1
- Integrals Class 12 Ex 7.1 in Hindi Medium
- Integrals Class 12 Ex 7.2
- Integrals Class 12 Ex 7.2 in Hindi Medium
- Integrals Class 12 Ex 7.3
- Integrals Class 12 Ex 7.3 in Hindi Medium
- Integrals Class 12 Ex 7.4
- Integrals Class 12 Ex 7.4 in Hindi Medium
- Integrals Class 12 Ex 7.5
- Integrals Class 12 Ex 7.5 in Hindi Medium
- Integrals Class 12 Ex 7.6
- Integrals Class 12 Ex 7.6 in Hindi Medium
- Integrals Class 12 Ex 7.7
- Integrals Class 12 Ex 7.7 in Hindi Medium
- Integrals Class 12 Ex 7.8
- Integrals Class 12 Ex 7.8 in Hindi Medium
- Integrals Class 12 Ex 7.9
- Integrals Class 12 Ex 7.9 in Hindi Medium
- Integrals Class 12 Ex 7.10
- Integrals Class 12 Ex 7.10 in Hindi Medium
- Integrals Class 12 Ex 7.11
- Integrals Class 12 Ex 7.11 in Hindi Medium
- Integrals Class 12 Miscellaneous Exercise
- RD Sharma Class 12 Integrals
- NCERT Exemplar Class 12 Maths Integrals
- Integrals Class 12 Notes Maths Chapter 7
- Important Questions Class 12 Maths Integrals
- JEE Main Mathematics Definite and Indefinite Integrals

### Class 12 Maths Chapter 7 NCERT Solutions Integrals

NCERT Solutions for Class 12 Maths Chapter 7 – Integrals contains step-by-step and detailed solutions for every question.

- Introduction
- Integration as an Inverse Process of Differentiation
- Geometrical interpretation of indefinite integral
- Some properties of indefinite integral
- Comparison between differentiation and integration
- Methods of Integration
- Integration by substitution
- Integration using trigonometric identities
- Integrals of Some Particular Functions
- Integration by Partial Fractions
- Integration by Parts
- Integral of the type
- Integrals of some more types
- Definite Integral
- Definite integral as the limit of a sum
- Fundamental Theorem of Calculus
- Area function
- First fundamental theorem of integral calculus
- Second fundamental theorem of integral calculus
- Evaluation of Definite Integrals by Substitution
- Some Properties of Definite Integrals

### Integrals Ex 7.1 Class 12

Question 1.

sin 2x

Solution:

Question 2.

cos 3x

Solution:

Question 3.

Solution:

Question 4.

(ax + c)²

Solution:

Question 5.

Solution:

Find the following integrals in Exercises 6 to 20 :

Question 6.

Solution:

Question 7.

Solution:

Question 8.

Solution:

Question 9.

Solution:

Question 10.

Solution:

Question 11.

Solution:

Question 12.

Solution:

Question 13.

Solution:

Question 14.

Solution:

Question 15.

Solution:

Question 16.

Solution:

Question 17.

Solution:

Question 18.

Solution:

Question 19.

Solution:

Question 20.

Solution:

Choose the correct answer in Exercises 21 and 22.

Question 21.

The antiderivative equals

(a)

(b)

(c)

(d)

Solution:

Question 22.

If such that f(2)=0 then f(x) is

(a)

(b)

(c)

(d)

Solution:

Move to Top → Class 12 Integrals Chapter 7

### Integrals Ex 7.2 Class 12

Integrate the functions in Exercises 1 to 37:

Question 1.

Solution:

Question 2.

Solution:

Question 3.

Solution:

Question 4.

sinx sin(cosx)

Solution:

Question 5.

sin(ax+b) cos(ax+b)

Solution:

Question 6.

Solution:

Question 7.

Solution:

Question 8.

Solution:

Question 9.

Solution:

Question 10.

Solution:

Question 11.

Solution:

Question 12.

Solution:

Question 13.

Solution:

Question 14.

Solution:

Question 15.

Solution:

Question 16.

Solution:

Question 17.

Solution:

Question 18.

Solution:

Question 19.

Solution:

Question 20.

Solution:

Question 21.

tan²(2x-3)

Solution:

∫tan²(2x-3)dx = ∫[sec²(2x-3)-1]dx = I

put 2x-3 = t

so that 2dx = dt

I = ∫sec²t dt-x+c

=

=

Question 22.

sec²(7-4x)

Solution:

Question 23.

Solution:

Question 24.

Solution:

Question 25.

Solution:

Question 26.

Solution:

Question 27.

Solution:

Question 28.

Solution:

Question 29.

cotx log sinx

Solution:

Question 30.

Solution:

Question 31.

Solution:

Question 32.

Solution:

Question 33.

Solution:

Question 34.

Solution:

Question 35.

Solution:

Question 36.

Solution:

Question 37.

Solution:

Choose the correct answer in exercises 38 and 39

Question 38.

(a) 10x – x10 + C

(b) 10x + x10 + C

(c) (10x – x10) + C

(d) log (10x + x10) + C

Solution:

Question 39.

(a) tanx + cotx + c

(b) tanx – cotx + c

(c) tanx cotx + c

(d) tanx – cot2x + c

Solution:

Move to Top → Class 12 Integrals Chapter 7

### Integrals Ex 7.3 Class 12

**Find the integrals of the functions in Exercises 1 to 22.**

Question 1.

sin²(2x+5)

Solution:

Question 2.

sin3x cos4x

Solution:

Question 3.

∫cos2x cos4x cos6x dx

Solution:

Question 4.

∫sin^{3}(2x+1)dx

Solution:

Question 5.

sin^{3}x cos^{3}x

Solution:

Question 6.

sinx sin2x sin3x

Solution:

Question 7.

sin 4x sin 8x

Solution:

Question 8.

Solution:

Question 9.

Solution:

Question 10.

∫sinx^{4} dx

Solution:

Question 11.

cos^{4} 2x

Solution:

Question 12.

Solution:

Question 13.

Solution:

Question 14.

Solution:

Question 15.

Solution:

Question 16.

tan^{4}x

Solution:

Question 17.

Solution:

Question 18.

Solution:

Question 19.

Solution:

Question 20.

Solution:

Question 21.

sin^{-1} (cos x)

Solution:

Question 22.

Solution:

Question 23.

(a) tanx+cotx+c

(b) tanx+cosecx+c

(c) -tanx+cotx+c

(d) tanx+secx+c

Solution:

Question 24.

(a) -cot(e.x^{x})+c

(b) tan(xe^{x})+c

(c) tan(e^{x})+c

(d) cot e^{x}+c

Solution:

Move to Top → Class 12 Integrals Chapter 7

### Integrals Class 12 Ex 7.4

Question 1.

Solution:

Question 2.

Solution:

Question 3.

Solution:

Question 4.

Solution:

Question 5.

Solution:

Question 6.

Solution:

Question 7.

Solution:

Question 8.

Solution:

Question 9.

Solution:

Question 10.

Solution:

Question 11.

Solution:

Question 12.

Solution:

Question 13.

Solution:

Question 14.

Solution:

Question 15.

Solution:

Question 16.

Solution:

Question 17.

Solution:

Question 18.

Solution:

put 5x-2=A(1+2x+3x²)+B

⇒ 6A=5, A=, B=

Question 19.

Solution:

Question 20.

Solution:

Question 21.

Solution:

Question 22.

Solution:

Question 23.

Solution:

Question 24.

(a) xtan^{-1}(x+1)+c

(b) (x+1)tan^{-1}x+c

(c) tan^{-1}(x+1)+c

(d) tan^{-1}x+c

Solution:

Question 25.

(a)

(b)

(c)

(d)

Solution:

Move to Top → Class 12 Integrals Chapter 7

### Integrals Class 12 Ex 7.5

Question 1.

Solution:

Question 2.

Solution:

Question 3.

Solution:

Question 4.

Solution:

Question 5.

Solution:

Question 6.

Solution:

Question 7.

Solution:

Question 8.

Solution:

Question 9.

Solution:

Question 10.

Solution:

Question 11.

Solution:

Question 12.

Solution:

Question 13.

Solution:

Question 14.

Solution:

Question 15.

Solution:

Question 16.

[Hint : multiply numerator and denominator by x^{n-1} and put x^{n} = t ]

Solution:

Question 17.

Solution:

Question 18.

Solution:

Question 19.

Solution:

Question 20.

Solution:

Question 21.

Solution:

Question 22.

choose the correct answer in each of the following :

(a)

(b)

(c)

(d) log|(x-1)(x-2)|+c

Solution:

Question 23.

(a)

(b)

(c)

(d)

Solution:

Move to Top → Class 12 Integrals Chapter 7

### Integrals Class 12 Ex 7.6

Question 1.

x sinx

Solution:

Question 2.

x sin3x

Solution:

Question 3.

Solution:

Question 4.

x logx

Solution:

Question 5.

x log2x

Solution:

Question 6.

Solution:

Question 7.

Solution:

Question 8.

Solution:

Question 9.

Solution:

Question 10.

Solution:

Question 11.

Solution:

Question 12.

x sec²x

Solution:

Question 13.

Solution:

Question 14.

x(logx)²

Solution:

Question 15.

(x²+1)logx

Solution:

Question 16.

Solution:

Question 17.

Solution:

Question 18.

Solution:

Question 19.

Solution:

Question 20.

Solution:

Question 21.

Solution:

Question 22.

Solution:

Choose the correct answer in exercise 23 and 24

Question 23.

(a)

(b)

(c)

(d)

Solution:

Question 24.

(a)

(b)

(c)

(d)

Solution:

Move to Top → Class 12 Integrals Chapter 7

### Class 12 Integrals Ex 7.7

Integral the function in exercises 1 to 9

Question 1.

Solution:

Question 2.

Solution:

Question 3.

Solution:

Question 4.

Solution:

Question 5.

Solution:

Question 6.

Solution:

Question 7.

Solution:

Question 8.

Solution:

Question 9.

Solution:

Choose the correct answer in the Exercises 10 to 11:

Question 10.

(a)

(b)

(c)

(d)

Solution:

Question 11.

Solution:

Move to Top → Class 12 Integrals Chapter 7

### Class 12 Integrals Ex 7.8

Question 1.

Solution:

Question 2.

Solution:

Question 3.

Solution:

Question 4.

Solution:

Question 5.

Solution:

Question 6.

Solution:

Move to Top → Class 12 Integrals Chapter 7

### Class 12 Integrals Ex 7.9

Question 1.

Solution:

Question 2.

Solution:

Question 3.

Solution:

Question 4.

Solution:

Question 5.

Solution:

Question 6.

Solution:

Question 7.

Solution:

Question 8.

Solution:

Question 9.

Solution:

Question 10.

Solution:

Question 11.

Solution:

Question 12.

Solution:

Question 13.

Solution:

Question 14.

Solution:

Question 15.

Solution:

Question 16.

Solution:

Question 17.

Solution:

Question 18.

Solution:

Question 19.

Solution:

Question 20.

Solution:

Question 21.

(a)

(b)

(c)

(d)

Solution:

Question 22.

(a)

(b)

(c)

(d)

Solution:

Move to Top → Class 12 Integrals Chapter 7

### Integration Class 12 Ex 7.10

Question 1:

Solution:

Question 2:

Solution:

Question 3:

Solution:

Question 4:

Solution:

Question 5:

Solution:

Question 6:

Solution:

Question 7:

Solution:

Question 8:

Solution:

Question 9:

Solution:

Question 10:

Solution:

Question 11:

Solution:

Question 12:

Solution:

Question 13:

Solution:

Question 14:

Solution:

Question 15:

Solution:

Question 16:

Solution:

Question 17:

Solution:

Question 18:

Solution:

Question 19:

Solution:

Question 20:

Solution:

Move to Top → Class 12 Integrals Chapter 7

### NCERT Solutions for Class 2 Maths Integration Class 12 Ex 7.11

Move to Top → Class 12 Integrals Chapter 7

### NCERT Solutions for Class 12 Maths Chapter 7 Integrals Miscellaneous Exercise

Question 1:

Solution:

Question 2:

Solution:

Question 3:

Solution:

Question 4:

Solution:

Question 5:

Solution:

Question 6:

Solution:

Question 7:

Solution:

Question 8:

Solution:

Question 9:

Solution:

Question 10:

Solution:

Question 11:

Solution:

Question 12:

Solution:

Question 13:

Solution:

Question 14:

Solution:

Question 15:

Solution:

Question 16:

Solution:

Question 17:

Solution:

Question 18:

Solution:

Question 20:

Solution:

Question 21:

Solution:

Question 22:

Solution:

Question 23:

Solution:

Question 24:

Solution:

Question 25:

Solution:

Question 26:

Solution:

Question 27:

Solution:

Question 28:

Solution:

Question 29:

Solution:

Question 30:

Question 31:

Solution:

Question 32:

Solution:

Question 33:

Solution:

Question 34:

Solution:

Question 35:

Solution:

Question 36:

Solution:

Question 37:

Solution:

Question 38:

Solution:

Question 39:

Solution:

Question 40:

Solution:

Question 41:

Solution:

Question 42:

Solution:

Question 43:

Solution:

Question 44:

Solution:

Move to Top → Class 12 Integrals Chapter 7

### Integration Formulas

Move to Top → Class 12 Integrals Chapter 7

### गणित कक्षा 12 समाकलन प्रश्नावली 7.1

निम्नलिखित फलनों के प्रतिअवकलज ( समाकलन) निरीक्षण विधि द्वारा ज्ञात कीजिए।

प्रश्न 1.

sin 2x

हल-

∫sin 2x dx

हम जानते हैं कि, cos2x = -2 sin 2x

प्रश्न 2.

cos 3x

हल-

∫cos 3x dx

हम जानते हैं कि, sin 3x = 3 cos 3x

प्रश्न 3.

e^{2x}

हल-

∫e^{2x} dx

हम जानते हैं कि,

प्रश्न 4.

(ax + b)²

हल-

∫(ax + b)² dx

हम जानते हैं कि,

प्रश्न 5.

sin 2x – 4e^{3x}

हल-

∫(sin 2x -4e^{3x}) dx = ∫sin 2x dx – 4∫e^{3x} …(1)

निम्नलिखित समाकलनों को ज्ञात कीजिए।

प्रश्न 6.

∫(4e^{3x} + 1) dx

हल-

∫(4e^{3x} + 1) dx = 4∫e^{3x}dx + ∫1 dx

प्रश्न 7.

हल-

प्रश्न 8.

∫(ax² + bx + c) dx

हल-

∫(ax² + bx + c) dx = a∫x²dx + b∫xdx + c∫1 dx

प्रश्न 9.

∫(2x² + e^{x}) dx

हल-

∫(2x² + e^{x}) dx = 2∫x² dx + ∫e^{x} dx

प्रश्न 10.

हल-

प्रश्न 11.

हल-

प्रश्न 12.

हल-

प्रश्न 13.

हल-

प्रश्न 14.

∫(1-x)√x dx

हल-

∫(1-x)√x dx

=∫(√x-x√x)dx

प्रश्न 15.

∫√x(3x² + 2x + 3)dx

हल-

∫√x(3x² + 2x + 3)dx

प्रश्न 16.

∫(2x – 3cosx + e^{x})dx

हल-

∫(2x – 3cosx + e^{x})dx

प्रश्न 17.

∫(2x² – 3sinx + 5√x)dx

हल-

∫(2x² – 3sinx + 5√x)dx

= 2∫x² dx – 3 ∫ sinx dx + 5∫√x dx

प्रश्न 18.

∫secx (sec x + tan x) dx

हल-

∫sec x (sec x + tan x) dx

= ∫(sec² x + sec x tan x) dx

= ∫(sec² x dx

= ∫sec x tan x dx

tan x + sec x + C

प्रश्न 19.

हल-

प्रश्न 20.

हल-

प्रश्न 21 एवं 22 में सही उत्तर का चयन कीजिए

प्रश्न 21

हल-

प्रश्न 22

हल-

Move to Top → Class 12 Integrals Chapter 7

### Class 12 Maths NCERT Solutions

**Chapter 1 Relations and Functions****Chapter 2 Inverse Trigonometric Functions****Chapter 3 Matrices****Chapter 4 Determinants****Chapter 5 Continuity and Differentiability****Chapter 6 Application of Derivatives****Chapter 7 Integrals Ex 7.1****Chapter 8 Application of Integrals****Chapter 9 Differential Equations****Chapter 10 Vector Algebra****Chapter 11 Three Dimensional Geometry****Chapter 12 Linear Programming****Chapter 13 Probability Ex 13.1**

Updated NCERT Solutions for Class 12 Maths for 2019 Exams.

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