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
(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
⇒ - 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
=
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
⇒ ad x = dt
∴ dx = (1/a) dt
7. Integrate the functions x√(x + 2)
Solution
Let (x + 2) = t
dx = dt
dx = dt
8. Integrate the functions x√(1 + 2x2 )
Solution
Let 1 + 2x2 = t
4x dx = dt
4x dx = dt
9. Integrate the functions (4x + 2) √(x2 + x + 1)
Solution
Let x2 + x + 1 = t
(2x + 1)dx = dt
(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
dx = dt
12. Integrate the functions (x3 - 1)1/3 x5
Solution
Let x3 - 1 = t
⇒ 3x2 dx = dt
⇒ 3x2 dx = dt
13. Integrate the functions x2/(2 + 3x3)3
Solution
Let 2 + 3x3 = t
⇒ 9x2 dx = dt
⇒ 9x2 dx = dt
14. Integrate the functions 1/x(log x)m, x > 0
Solution
Let log x = t
Solution
Let 9 - 4x2 = t
-8x dx = dt
-8x dx = dt
16. Integrate the functions e2x+3
Solution
Let 2x + 3 = t
2dx = dt
2dx = dt
17. Integrate the functions x/ex2
Solution
Let x2 = t
2x dx = dt
Let 3 cos x + 2 sin x = t
(-3 sin x + 2cos x) dx = dt
Let (1 - tan x) = t
⇒ - sec2 x dx = dt
= 2 sin t + C
= 2 sin √x + C
= - |log|t| + C
= - log|1 + cos x| + C
Let sin x + cos x = t then (cos x - sin x) dx = dt
Put cos x - sin x = t then (- sin x - cos x) dx = dt
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
Dividing numerator and denominator by ex , we obtain
Let ex + e-x = t
⇒ (ex - e-x )dx = dt
⇒ (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
⇒ (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
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
-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
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
⇒ 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
⇒ (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
⇒ - 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
⇒ - 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
Let tan-1 t = u
⇒ 1/(1 + t2) dt = du
From (1), we obtain
Hence, the correct answer is D.
⇒ (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
⇒ 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
(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
(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.