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

6. Integrate the functions √(ax + b)
Solution
Let ax + b = t
∴ 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.