NCERT Solutions class 12 Maths Exercise 7.11

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 =

=

=

=

=

=

=

---

By using the properties of definite integrals, evaluate the integrals in Exercises 7 to 11.

7.

Ans. Let I =

=

 I =

=

=

 I =

=

=

---

8.

Ans. Let I =         ……….(i)

 I =

 I =

=

=  …..(ii)

Adding eq. (i) and (ii),

2I =

=

 2I =

=

=

 I =

---

9.  

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

---

14.

Ans.

=

Here

 

 

---

NCERT Solutions class 12 Maths Exercise 7.11

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

---

NCERT Solutions class 12 Maths Exercise 7.11

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.

---

NCERT Solutions class 12 Maths Exercise 7.11

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.

---

NCERT Solutions class 12 Maths Exercise 7.11

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

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