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# NCERT Solutions class 12 Maths Exercise 7.11

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NCERT Solutions class 12 Maths Exercise 7.11 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.

Download NCERT solutions for Integrals as PDF.

## NCERT Solutions class 12 Maths Integrals

By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 6.

1.

Ans. Let I =  ……….(i)

=

……….(ii)

Adding eq. (i) and (ii),

2I =

=

=

2I =

I =

#### 2.

Ans. Let I =   ………..(i)

I =

I =  ……….(ii)

Adding eq. (i) and (ii),

2I =

=

2I =  =

2I =

I =

#### 3.

Ans. Let I =   ………..(i)

I =

I =   ……….(ii)

Adding eq. (i) and (ii),

2I =

=

2I =  =

2I =

I =

#### 4.

Ans. Let I =  ………..(i)

I =

I =  ……….(ii)

Adding eq. (i) and (ii),

2I =

=

2I =  =

2I =    I =

#### 5.

Ans. Let I =  ……….(i)

Putting

From eq. (i),

I =

=

=

=

=

=

=

#### 6.

Ans. Let I =   ……….(i)

Putting

From eq. (i),

I =

=

=

=

=

=

=

7.

Ans. Let I =

=

I =

=

=

I =

=

=

#### 8.

Ans. Let I =         ……….(i)

I =

I =

=

=  …..(ii)

Adding eq. (i) and (ii),

2I =

=

2I =

=

=

I =

Ans. Let I =

=

I =

=

=

=

I =

=

=

#### 10.

Ans. Let I =

=

=

=

=  ……….(i)

I =

I =         ……….(ii)

Adding eq. (i) and (ii),

2I =

=

2I =

=

=

=

I =

=

=

#### 11.

Ans. Let I =

= …(i)

I =

I =         ……….(ii)

Adding eq. (i) and (ii),

2I =

=

=

=

I =

##### Using properties of definite integrals, evaluate the following integrals in Exercise 12 to 18.

12.

Ans. Let I =          ……….(i)

I =

=      ……….(ii)

Adding eq. (i) and (ii),

2I =

=

=

=

2I =

2I =

2I =

I =

=

=

=

=

#### 13.

Ans. Let I =

Here

=

=

is an odd function of

I =  = 0

Ans.

=

Here

### 15.

Ans. Let I =  ……….(i)

I =

=

=  ….(ii)

Adding eq. (i) and (ii), we have 2I = 0   I = 0

### 16.

Ans. Let I =         ……….(i)

I =

=     ……….(ii)

Adding eq. (i) and (ii),

2I =

=

2I =

=

=

I =  ……….(iii)

I =

=      ……….(iv)

Adding eq. (i) and (ii),

2I =

=

I =

=

=

I =

=

=

I = I1 ……….(v)

Where I1 =  …….(vi)

Putting  in eq. (vi),

##### Limits of integration when  and

From eq. (vi),

I1 =

=

=

I1 =

=

I1 =  [From eq. (iii)]

Putting this value in eq. (v), I =

2I =

2I – I =

I =

#### 17.

Ans. Let I =  ……….(i)

I =

=     ……….(ii)

Adding eq. (i) and (ii),

2I =

=

=  =  =

I =

#### 18.

Ans. Let I =          ……….(i)

Here

From eq. (i),

I =

=

I =

=

I =

=

= 6 – 1

= 5

### 19. Show that  are defined as  and

Ans. Here  ……..(i)  and   ………(ii)

Let I =  ……….(iii)

I =

=  [From eq. (i)] ……….(iv)

Adding eq. (iii) and (iv)

2I =

=

2I =

[From eq. (ii)]

2I =

I =

Hence proved.

### Choose the correct answer in Exercises 20 and 21.

20. The value of  is:

(A) 0

(B) 2

(C)

(D) 1

Ans. Let I =

=

I =

=

Therefore, option (C) is correct.

### 21. The value of  is:

(A) 2

(B)

(C) 0

(D)

Ans. Let I =  ……….(i)

I =

=  ……….(ii)

Adding eq. (i) and (ii),

2I =

=

2I =

=  = 0

I = 0

## NCERT Solutions class 12 Maths Exercise 7.11

NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 13 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide

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### 8 thoughts on “NCERT Solutions class 12 Maths Exercise 7.11”

1. Yo

2. Very useful , and understandable step wise solution helps students to understand easily .

3. Vow yar what’s notes

4. Very good problems for the students to
Learn better and understand easily

5. Nice teaching

6. good solution

7. WELL….IMPRESSIVE?

8. best solutions