NCERT solution class 12 chapter 1 Integrals exercise 1.11 mathematics part 2

EXERCISE 1.11

Question 1:

Adding (1) and (2), we obtain

Question 2:

Adding (1) and (2), we obtain

Question 3:

Adding (1) and (2), we obtain

Question 4:

Adding (1) and (2), we obtain

Question 5:

It can be seen that (x + 2) ≤ 0 on [−5, −2] and (x + 2) ≥ 0 on [−2, 5].

Question 6:

It can be seen that (x − 5) ≤ 0 on [2, 5] and (x − 5) ≥ 0 on [5, 8].

Question 10:

Adding (1) and (2), we obtain

Question 11:

As sin(−x) = (sin (−x))2 = (−sin x)2 = sin2x, therefore, sin2is an even function.

It is known that if f(x) is an even function, then

Question 12:

Adding (1) and (2), we obtain

Question 13:

As sin(−x) = (sin (−x))7 = (−sin x)7 = −sin7x, therefore, sin2is an odd function.

It is known that, if f(x) is an odd function, then

Question 14:

It is known that,

Question 15:

Adding (1) and (2), we obtain

Question 16:

Adding (1) and (2), we obtain

sin (π − x) = sin x

Adding (4) and (5), we obtain

Let 2x = t ⇒ 2dx = dt

When x = 0, = 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:

It is known that,

Adding (1) and (2), we obtain

Question 18:

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

Adding (1) and (2), we obtain

Question 20:

The value of is

A. 0

B. 2

C. π

D. 1

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.

The value of is

A. 2

B.

C. 0

D.