Class 12 Maths NCERT Solutions for Chapter 7 Integrals 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 ∫[ex (1 + x)]/[cos2 (ex x)] dx
(A) -cot (ex x) + C
(B) tan (xex ) + C
(C) tan (ex ) + C
(D) cot(ex ) + C
Solution

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

Hence, the correct answer is B.
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