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Class 12 Maths NCERT Solutions for Chapter 7 Integrals Exercise 7.2

Class 12 Maths NCERT Solutions for Chapter 7 Integrals Exercise 7.2

Integrals Exercise 7.2 Solutions

1. Integrate the functions 2x/(1 + x2 ).

Solution

Let 1 + x2 = t 
2x dx = dt 

= log |t| + C
= log|1 + x2 | + C 
= log(1 + x2 ) + C


2. Integrate the functions (log x)2 /x 

Solution 

Let log |x| = t 
(1/x) dx = dt 


3. Integrate the functions 1/(x + x log x) 

Solution

Let 1 + log x = t 
(1/x) dx = dt 

= log |t| + C 
= log |1 + log x| + C 

4. Integrate the functions sin x ⋅ sin (cos x) .
Solution
Let cos x = t 
⇒ - sin x dx = dt 
⇒ ∫ sin x . sin(cos x) dx = - ∫sin t dt 
= - [-cos t] + C  
= cos t + C 
= cos (cos x) + C 

5. Integrate the functions sin (ax + b) cos (ax + b) 
Solution
sin (ax + b) cos (ax + b) 

Let 2(ax + b) = t 
2adx = dt 

6. Integrate the functions √(ax + b)
Solution
Let ax + b = t 
⇒ ad x = dt 
∴ dx = (1/a) dt 

7. Integrate the functions x√(x + 2) 
Solution
Let (x + 2) = t 
dx = dt 

8. Integrate the functions x√(1 + 2x2 )
Solution
Let  1 + 2x2 = t 
4x dx = dt 

9. Integrate the functions (4x + 2) √(x2 + x + 1) 
Solution
Let x2 + x + 1 = t 
(2x + 1)dx = dt 

10. Integrate the functions 1/(x - √x) .
Solution

11. Integrate the functions x/√(x + 4) , x > 0 
Solution
Let x + 4  = t 
dx = dt

12. Integrate the functions (x3 - 1)1/3 x5 
Solution
Let x3 - 1 = t 
⇒ 3x2 dx = dt 

13. Integrate the functions x2/(2 + 3x3)3 
Solution
Let 2 + 3x3 = t 
⇒ 9x2 dx = dt 

14. Integrate the functions 1/x(log x)m, x > 0
Solution
Let log x = t 

15. Integrate the functions x/(9 - 4x2 ) 
Solution
Let 9 - 4x2 = t 
-8x dx = dt 

16. Integrate the functions e2x+3 
Solution
Let 2x + 3 = t
2dx = dt 

17. Integrate the functions x/ex2
Solution
Let x2 = t 
2x dx = dt 

18. Integrate the functions etan-1 x /(1 + x2 )
Solution
Let tan-1 x = t


19. Integrate the functions (e2x - 1)/(e2x + 1)
Solution
(e2x - 1)/(e2x + 1)
Dividing numerator and denominator by ex , we obtain 
 Let ex + e-x  = t 
⇒ (ex - e-x )dx = dt 

20. Integrate the functions (e2x - e-2x )/(e2x + e-2x )
Solution
Let e2x + e-2x = t 
⇒ (2e2x - 2e-2x)dx = dt 
⇒ 2(e2x - e-2x) dx = dt 

21. Integrate the functions tan2 (2x - 3)
Solution
tan2 (2x - 3) = sec2 (2x - 3) - 1 
Let 2x - 3 = t 
⇒ 2dx = dt 
⇒ ∫tan2 (2x - 3)dx 

22. Integrate the functions sec2 (7 - 4x) 
Solution
Let 7 - 4x = t 
-4dx = dt 
∴ ∫sec2 (7 - 4x)dx 

23. Integrate the functions (sin-1 x)/√(1 - x2 )
Solution
Let sin-1 x = t 

24. Integrate the functions (2 cos x - 3 sin x)/(6 cos x + 4 sin x)
Solution 

Let 3 cos x + 2 sin x = t 
(-3 sin x + 2cos x) dx = dt 

25. Integrate the functions 1/[cos23 x(1 - tan x)2 ]
Solution

Let (1 - tan x) = t 
⇒ - sec2 x dx = dt 

26. Integrate the functions cos√x/√x
Solution
Let √x = t 

= 2 sin t + C 
= 2 sin √x + C 

27. Integrate the functions √(sin 2x) cos 2x 
Solution
Let sin 2x = t 
2 cos 2x dx = dt 
⇒ ∫√(sin 2x) cos 2x dx = (1/2)∫√t dt 

28. Integrate the functions cos x/√(1 + sin x) .
Solution
Let 1 + sin x = t 
⇒ cos x dx = dt 

29. Integrate the function cot x log sin x
Solution
Let log sin x = t 
⇒ (1/sin x) . cos x dx = dt 
∴ cot x dx = dt 
∫ cot x log sin x dx = ∫t dt 

30. Integrate the functions sinx/(1 + cos x) 
Solution
Let 1 + cos x = t 
⇒ - sin x dx = dt 

= - |log|t| + C 
= - log|1 + cos x| + C 

31. Integrate the functions in sin x/(1 + cos x)2 
Solution 
Let 1 + cos x = t 
⇒ - sin x dx = dt 

32. Integrate the functions in 1/(1 + cot x) 
Solution

Let sin x + cos x = t then (cos x - sin x) dx = dt 

33. Integrate the functions in 1/(1 - tan x) 
Solution

Put cos x - sin x = t  then (- sin x - cos x) dx = dt 

34. Integrate the functions in √(tan x)/(sin x cos x) . 
Solution

35. Integrate the functions in (1 + log x)2/x
Solution
Let 1 + log x = t 
⇒ (1/x) dx = dt 

36. Integrate the function in[(x + 1)(x + log x)2 /x 
Solution

37. Integrate the function in x3 sin (tan-1 x4)/(1 + x8
Solution
Let x4 = t 
⇒ 4x3 dx = dt 

Let tan-1 t = u 
⇒ 1/(1 + t2) dt = du 
From (1), we obtain 

38. Choose the correct answer int (10x9 + 10x loge 10)/ (x10 + 10x) dx equals 
(A) 10x - x10 + C 
(B) 10x + x10 + C 
(C) (10x - x10) - 1 + C
(D) log (10x + x10) + C 
Solution
Let x10 + 10x = t 

Hence, the correct answer is D. 

39. Choose the correct answer ∫dx/(sin2 x cos2 x) equals 
(A) tan x + cot x + C 
(B) tan x - cot x + C 
(C) tan x cot x + C 
(D) tan x - cot 2x + C 
Solution
Hence, the correct answer is B. 
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