NCERT Solutions for Chapter 7 Integrals Class 12 Maths Exercise 7.3

Class 12 Maths NCERT Solutions for Chapter 7 Integrals Exercise 7.3

Integrals Exercise 7.3 Solutions

1. Find the integrals of the functions sin² (2x + 5)

Solution

2. Find the integrals of the functions sin 3x cos 4x.

Solution

It is known that, sin A cos B = (1/2) {sin (A + B) + sin (A - B)} 

3. Find the integrals of the functions cos 2x cos 4x cos 6x . 

Solution 

It is known that, cos A cos B = (1/2) {cos (A + B) + cos (A - B)} 

 

4. Find the integrals of the functions sin³ (2x + 1).

Solution 

Let I  = ∫ sin³ (2x + 1) 
⇒  ∫sin³ (2x + 1) dx = ∫ sin² (2x + 1) . sin(2x + 1) dx 
= ∫[1 - cos² (2x + 1)] sin(2x + 1)dx

Let cos(2x + 1) = t 
⇒ -2sin(2x + 1)dx = dt 
⇒ sin(2x + 1)dx = -dt/2 

5. Find the integrals of the functions sin³x cos³x

Solution 

Let I = ∫ sin³ x cos³ x dx 
= ∫ cos³ x . sin² x . sin x . dx 
= ∫ cos³ x(1 - cos² x) sin x . dx 
Let cos x = t 
⇒ - sin x. dx = dt 

6. Find the integrals of the functions sin x sin 2x sin 3x 

Solution

It is known that, sin A sin B = (1/2) {cos (A - B) - cos (A + B)} 

7. Find the integrals of the functions sin 4x sin 8x.

Solution

It is known that, sin A sin B = (1/2) {cos (A - B) - cos (A + B)} 

8. Find the integrals of the functions (1 - cos x)/(1 + cos x) 

Solution

9. Find the integrals of the functions cos x/(1 + cos x).

Solution

10. Find the integrals of the functions sin⁴ x 

Solution

sin⁴ x = sin² x sin² x 

11. Find the integrals of the functions cos⁴ 2x 

Solution

12. Find the integrals of the functions sin² x/(1 + cos x)

Solution

13. Find the integrals of the functions (cos 2x - cos 2α)/(cos x - cos α)

Solution

14. Find the integrals of the functions (cos x - sin x)/(1 + sin 2x)

Solution

15. Find the integrals of the functions tan³ 2x sec 2x.

Solution

tan³2x sec 2x = tan² 2x tan 2x sec 2x 
= (sec² 2x - 1) tan 2x sec 2x 
= sec² 2x. tan 2x sec 2x - tan 2x sec2x 
  ∴ ∫tan³ 2x sec 2x dx = ∫ sec² 2x tan 2x sec 2x dx - ∫ tan 2x sec 2x dx 
= ∫ sec² 2x tan 2x sec 2x dx - sec 2x/2 + C 
Let sec 2x = t 
∴ 2 sec 2x tan 2x dx = dt 

16. Find the integrals of the functions tan⁴x . 

Solution

tan⁴ x 
= tan² x . tan² x 
= (sec² x - 1) tan² x 
= sec² x tan² x - tan² x 
= sec² x tan² x - (sec² x - 1) 
= sec² x tan² x - sec² x + 1 

∴ ∫ tan⁴ x dx = ∫ sec² x tan² x dx - ∫ sec² x dx + ∫ 1 . dx 
= ∫ sec² x tan² x dx - tan x + x + C ...(1) 
Consider ∫ sec² x tan² x dx 
Let tan x = t ⇒ sec² x dx = dt 
⇒ ∫ sec² x tan² x dx = ∫t² dt = t³/3 = tan³ x/3 

From equation (1), we obtain 
∫ tan⁴ x dx = (1/3) tan³ x - tan x + x + C 

17. Find the integrals of the functions (sin³ x + cos³ x)/(sin² x cos² x) . 

Solution

18. Find the integrals of the functions (cos 2x + 2 sin² x)/(cos² x) 

Solution

19. Find the integrals of the functions 1/(sin x cos³ x) .

Solution

20. Find the integrals of the functions cos 2x/(cos x + sin x)² .

Solution

21. Find the integrals of the functions sin^-1 (cos x). 

Solution

sin^-1 (cos x) 
Let cos x = t 
Then, sin x = √(1 - t² ) 
⇒ (- sin x)dx = dt 

It is known that, 
sin^-1 x + cos^-1 x = π/2 

22. Find the integrals of the functions 1/[cos(x - a) cos(x - b)]  . 

Solution

23. Choose the correct answer ∫(sin² x - cos² x)/(sin² x cos² x) dx 
A. tan x + cot x + C
B. tan x + cosec x + C
C. −tan x + cot x + C
D. tan x + sec x + C

Solution

= tan x + cot x + C 
Hence, the correct answer is A. 

24. Choose the correct answer ∫[e^x (1 + x)]/[cos² (e^x x)] dx
(A) -cot (e^x x) + C
(B) tan (xe^x ) + C
(C) tan (e^x ) + C
(D) cot(e^x ) + C

Solution

Let e^x. x = t
⇒ (e^x. x + e^x.1) dx = dt 
e^x (x + 1)dx = dt 

Hence, the correct answer is B.