## NCERT Solutions for Class 12 Maths Chapter 7 – Integrals Ex 7.11

#### Page No 347:

#### Question 1:

#### Answer:

Adding (1) and (2), we obtain

#### Question 2:

#### Answer:

Adding (1) and (2), we obtain

#### Question 3:

#### Answer:

Adding (1) and (2), we obtain

#### Question 4:

#### Answer:

Adding (1) and (2), we obtain

#### Question 5:

#### Answer:

It can be seen that (

*x*+ 2) ≤ 0 on [−5, −2] and (*x*+ 2) ≥ 0 on [−2, 5].#### Question 6:

#### Answer:

It can be seen that (

*x*− 5) ≤ 0 on [2, 5] and (*x*− 5) ≥ 0 on [5, 8].#### Question 7:

#### Answer:

#### Question 8:

#### Answer:

#### Question 9:

#### Answer:

#### Question 10:

#### Answer:

Adding (1) and (2), we obtain

#### Question 11:

#### Answer:

As sin

^{2 }(−*x*) = (sin (−*x*))^{2}= (−sin*x*)^{2}= sin^{2}*x*, therefore, sin^{2}*x*is an even function.
It is known that if

*f*(*x*) is an even function, then#### Question 12:

#### Answer:

Adding (1) and (2), we obtain

#### Question 13:

#### Answer:

As sin

^{7 }(−*x*) = (sin (−*x*))^{7}= (−sin*x*)^{7}= −sin^{7}*x*, therefore, sin^{2}*x*is an odd function.
It is known that, if

*f*(*x*) is an odd function, then#### Question 14:

#### Answer:

It is known that,

#### Question 15:

#### Answer:

Adding (1) and (2), we obtain

#### Question 16:

#### Answer:

Adding (1) and (2), we obtain

sin (π −

*x*) = sin*x*
Adding (4) and (5), we obtain

Let 2

*x*=*t*⇒ 2*dx*=*dt*
When

*x*= 0,*t*= 0 and when
x=π2, t=π∴

I=12∫0πlog sin tdt-π2log 2

⇒I=I2-π2log 2 [from 3]

⇒I2=-π2log 2

⇒I=-πlog 2

#### Question 17:

#### Answer:

It is known that,

Adding (1) and (2), we obtain

#### Question 18:

#### Answer:

It can be seen that, (

*x*− 1) ≤ 0 when 0 ≤*x*≤ 1 and (*x*− 1) ≥ 0 when 1 ≤*x*≤ 4#### Question 19:

Show that if

*f*and*g*are defined as and#### Answer:

Adding (1) and (2), we obtain

#### Question 20:

The value of is

**A.**0

**B.**2

**C.**π

**D.**1

#### Answer:

It is known that if

*f*(*x*) is an even function, then and
if

*f*(*x*) is an odd function, then
Hence, the correct answer is C.

#### Question 21:

The value of is

**A.**2

**B.**

**C.**0

**D.**

#### Answer:

Adding (1) and (2), we obtain

Hence, the correct answer is C.

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