Question

NCERT Solutions Class 12 Maths Chapter 7 Integrals is one of the best study materials for the CBSE board term 1 exam. Our NCERT Solutions is designed by mentors who have years of experience in the field. NCERT Solutions Class 12 also covers various topics of maths such as integrals, integration, etc.

• Integrals – the integrals can be defined as the area under the curve region in a graph. A region bounded by a graph of any function calculated between two points is termed the definite integral of any given function. Intuitively we can see that the bounded region is further divided into thin rectangular areas with the help of lower and upper limits. Then the area is calculated by taking the algebraic sum of the area of the entire region.
• Integration as an Inverse Process of Differentiation – in the differentiation we calculate the derivative of any function while the integration is the process of calculating the antiderivative of a function. So we can say that integration is the inverse process of differentiation.
• Methods of Integration – the method of integration is simply a calculation of adding large values where the general method of algebraic operations is not possible. Hence there are various methods of integration. It is easier to calculate the original integral. There are other different methods of integration
  1. Integration by Substitution.
  2. Integration by Parts
  3. Integration Using Trigonometric Identities
  4. Integration of Some particular function
  5. Integration by Partial Fraction 
• Integration by Partial Fractions – with the help of integration of a partial fraction we can integrate a partial fraction to help integrate a rational fraction integrand that has complex terms of a denominator. We can decompose an expression into many simpler terms which can be used to easily calculate or integrate the expression.

NCERT Solutions Class 12 Maths has in-depth discussions and solutions to all sorts of questions in the related chapter.

Answer 1

NCERT Solutions Class 12 Maths Chapter 7 Integrals

1. Find an anti-derivative (or integral) of the following functions by the method of inspection. 

sin 2x

Answer:

The anti-derivative of sin 2x is a function of x whose derivative is sin 2x.

It is known that,

\(\frac d{dx}(cos2x) = -2sin 2x\)

⇒ \(sin\, 2x=-\frac12\frac d{dx}(cos \,2x)\)

\(\therefore sin\, 2x = \frac d {dx}\left(-\frac12 cos\,2x\right)\)

Therefore, the anti-derivative of sin 2x is \(-\frac12\)cos 2x.

2. Find an anti-derivative (or integral) of the following functions by the method of inspection. 

cos 3x

Answer:

The anti-derivative of cos 3x is a function of x whose derivative is cos 3x. It is known that,

\(\frac d{dx}(sin3x) = 3cos 3x\)

⇒ \(cos\, 3x=\frac13\frac d{dx}(sin \,3x)\)

\(\therefore cos\, 3x = \frac d {dx}\left(\frac13 sin\,3x\right)\)

Therefore, the anti-derivative of cos 2x is \(\frac13\)sin 2x.

3. Find an anti-derivative (or integral) of the following functions by the method of inspection. 

e^2x

Answer:

The anti-derivative of e^2x is the function of x whose derivative is e^2x. 

It is known that,

\(\frac d{dx}(e^{2x}) = 2e^{2x}\)

 ⇒ \(e^{2x} = \frac12\frac{d}{dx}(e^{2x})\)

\(\therefore e^{2x} = \frac{d}{dx}\left(\frac12 e^{2x}\right)\)

Therefore, the anti-derivative of e^2x is \(\frac13\)e^2x.

4. Find an anti-derivative (or integral) of the following functions by the method of inspection. 

(ax + b)²

Answer:

The anti-derivative of (ax + b)² is the function of x whose derivative is (ax + b)². 

It is known that,

\(\frac d{dx}(ax + b)^3 = 3a(ax + b)^2\)

⇒ \((ax + b)^2= \frac1{3a}\frac{d}{dx}(ax + b)^3\)

\(\therefore (ax + b)^2 = \frac{d}{dx}\left(\frac1{3a} (ax + b)^3\right)\)

Therefore, the anti derivative of (ax + b)² is \(\frac1{3a}\)(ax + b)³.

5. Find an anti-derivative (or integral) of the following functions by the method of inspection. 

sin 2x – 4e^3x

Answer:

The anti-derivative of sin 2x – 4e^3x is the function of x whose derivative is sin2x – 4e^3x 

It is known that,

\(\frac d{dx}\left(-\frac12 cos 2x -\frac43e^{3x}\right)= sin\,2x- 4e^{3x}\)

Therefore, the anti derivative of sin 2x – 4e^3x is \(\left(-\frac12 cos 2x -\frac43e^{3x}\right)\).

6. \(\int (4e^{3x} + 1)dx\) 

Answer:

\(\int (4e^{3x} + 1)dx\)

\(= 4\int e^{3x}dx + \int 1dx\)

\(= 4\left(\frac{e^{3x}}{3}\right)+ x +C\)

\(= \frac43e^{3x} + x +C\)

7. \(\int (ax^2 + bx + c)dx\) 

Answer:

\(\int (ax^2 + bx + c)dx\)

\(= a\int x^2 dx +b\int xdx + c \int1.dx\)

\(= a\left(\frac{x^3}3\right)+ b\left(\frac{x^2}{2}\right)+ cx +C\)

\(= \frac{ax^3}{3} + \frac{bx^2}{2}+ cx +C\)

8. \(\int x^2\left(1 - \frac{1}{x^2}\right)dx\) 

Answer:

\(\int x^2\left(1 - \frac{1}{x^2}\right)dx\) 

\(= \int (x^2 - 1)dx\)

\(= \int x^2dx - \int 1dx\)

\(= \frac{x^3}{3} - x + C\)

9. \(\int (2x^2 + e^x)dx\) 

Answer:

\(\int (2x^2 + e^x)dx\) 

\(= 2\int x^2 dx + \int e^x dx\)

\(= 2\left(\frac{x^3}{3}\right) + e^x + C\)

\(= \frac23 x^3 + e^x + C\) 

10. \(\int\left(\sqrt x - \frac 1 {\sqrt x}\right)^2 dx\) 

Answer:

\(\int\left(\sqrt x - \frac 1 {\sqrt x}\right)^2 dx\) 

\(= \int \left(x + \frac 1x -2\right)dx\)

\(= \int xdx + \int \frac 1x dx - 2 \int 1 .dx\)

\( = \frac{x^2 }{2} + log|x| - 2x + C\)

11. \(\int \frac{x^3 + 5x^2 -4}{x^2}dx\) 

Answer:

\(\int \frac{x^3 + 5x^2 -4}{x^2}dx\)

\(= \int (x +5 - 4x^{-2})dx\)

\( = \int xdx + 5 \int 1.dx - 4 \int x^{-2}dx\)

\(= \frac{x^2}{2} + 5x - 4 \left(\frac{x^{-1}}{-1}\right) + C \) 

\(= \frac{x^2 }{2 } + 5x + \frac 4x + C\)

12. \(\int \frac{x^3 + 3x +4}{\sqrt x}dx\) 

Answer:

\(\int \frac{x^3 + 3x +4}{\sqrt x}dx\)

\(= \int \left(x^{\frac 32} + 3x^{\frac12} + 4x ^{\frac 12}\right)dx\)

\(= \cfrac{x^{\frac72}}{\frac72} + \cfrac{3(x^{\frac32})}{\frac32} + \cfrac{4(x^{\frac12})}{\frac12} + C\)

\(= \frac27 x^{\frac72} + 2x^{\frac32} + 8x^{\frac12} + C\)

\(= \frac27 x^{\frac72} + 2x^{\frac32} + 8\sqrt x + C\)

13. \(\int \frac{x^3 - x^2 + x -1}{x -1}dx\) 

Answer:

\(\int \frac{x^3 - x^2 + x -1}{x -1}dx\) 

On dividing, we obtain

\(= \int (x^2 + 1 )dx\)

\( = \int x^2dx + \int 1 dx\)

\( = \frac{x^3}{3} + x+C\) 

14. \(\int (1 -x) \sqrt x dx\) 

Answer:

\(\int (1 -x) \sqrt x dx\) 

\(= \int (\sqrt x - x^{\frac 32})dx\)

\(= \int x^{\frac12} dx - \int x^{\frac32}dx\)

\( = \cfrac{x^{\frac32}}{\frac32} - \cfrac{x^{\frac52}}{\frac52}+ C\)

\( = \frac23 x^{\frac32} - \frac25 x^{\frac52} + C\)

Answer 2

15. \(\int \sqrt x (3x^2 + 2x + 3) dx\) 

Answer:

\(\int \sqrt x (3x^2 + 2x + 3) dx\) 

\( = \int \left(3x^{\frac52} + 2x^{\frac32} + 3x^{\frac12}\right)dx\)

\( = 3 \int x^{\frac52} dx + 2 \int x^{\frac32}dx + 3 \int x^{\frac12}dx\)

\( = 3\left(\frac{x^{\frac72}}{\frac72}\right) + 2 \left(\frac{x^{\frac52}}{\frac52}\right) + 3\frac{\left(x^{\frac32}\right)}{\frac32} + C\)

\( = \frac67 x^{\frac72} + \frac45 x ^{\frac52} + 2x^{\frac32} +C\) 

16. \(\int (2x - 3 cos x + e^x)dx\) 

Answer:

17. \(\int (2x^2 -3sinx + 5\sqrt x)dx\) 

Answer:

18. \(\int sec x (sec x + tan x)dx\) 

Answer:

19. \(\int\frac{sec^2x}{cosec^2 x}dx\) 

Answer:

20. \(\int \frac{2 - 3sinx}{cos ^2x}dx\) 

Answer: 

21. The anti-derivative of \(\left(\sqrt x + \frac1{\sqrt x}\right)\) equals

(A) \(\frac13 x^{\frac13} + 2x^{\frac12} + C\)

(B) \(\frac23 x^{\frac23} + \frac12 x^2 + C\)

(C) \(\frac23 x^{\frac32} + 2x^{\frac12} + C\)

(D) \(\frac32 x^{\frac32} + \frac12x^{\frac12} +C\)

Answer:

Hence, the correct Answer is C.

22. If \(\frac d{dx} f(x) = 4x^3 - \frac3{x^4}\) such that f(2) = 0, then f(x) is

(A) \(x^4 + \frac1{x^3} - \frac{129}8\)

(B) \(x^3 + \frac1{x^4} + \frac{129}8\)

(C) \(x^4 + \frac1{x^3} + \frac{129}8\)

(D) \(x^3 + \frac1{x^4} - \frac{129}8\)

Answer:

It is given that,

\(\frac d{dx} f(x) = 4x^3 - \frac3{x^4}\) 

Anti-derivative of \( 4x^3 - \frac3{x^4} = f(x)\) 

Also,

Hence, the correct Answer is A.

23. Integrate the function:

\(\frac{2x}{1 +x^2}\)

Answer:

Let 1 + x² = t 

\(\therefore\) 2x dx = dt

24. Integrate the function:

\(\frac{(log\,x)^2}{x}\) 

Answer:

Let log |x| = t

25. Integrate the function:

\(\frac{1}{x + x\,log\,x}\) 

Answer:

\(\frac{1}{x + x\,log\,x} = \frac1{x( 1 + log\,x)}\)

Let 1 + log x = t

26. Integrate the function:

sin x ⋅ sin (cos x)

Answer:

sin x ⋅ sin (cos x) 

Let cos x = t 

∴ −sin x dx = dt

27. Integrate the function:

\(sin(ax + b) cos (ax + b)\)

Answer:

28. Integrate the function:

\(\sqrt{ax + b}\)

Answer:

Let ax + b = t 

⇒ adx = dt

29. Integrate the function:

\(x\sqrt{x + 2}\)

Answer:

Let x + 2 = t 

\(\therefore \) dx = dt

30. Integrate the function:

\(x\sqrt{1 + 2x^2 }\)

Answer:

Let 1 + 2x² = t 

\(\therefore \) 4xdx = dt

31. Integrate the function:

\((4x + 2) \sqrt{x^2 + x + 1}\)

Answer:

Let x² + x + 1 = t 

\(\therefore \) (2x + 1)dx = dt

32. Integrate the function:

\(\frac1{x - \sqrt x}\)

Answer:

33. Integrate the function:

\(\frac x{\sqrt{x + 4}}, x>0\)

Answer:

Let x + 4 = t 

\(\therefore \) dx = dt

Answer 3

34. Integrate the function:

\((x^3 - 1)^\frac13 x^5\)

Answer:

Let x³ – 1 = t

 \(\therefore \) 3x²dx = dt

35. Integrate the function:

\(\frac{x^2}{(2 + 3x^3)^3}\)

Answer:

Let 2 + 3x³ = t 

\(\therefore \) 9x² dx = dt

36. Integrate the function:

\(\frac1{x(log\,x)^m}, x>0\)

Answer:

Let log x = t

\(\therefore \) \(\frac1x\) dx = dt

37. Integrate the function:

\(\frac x{9 - 4x^2}\) 

Answer:

Let 9 – 4x² = t 

\(\therefore \) −8x dx = dt

38. Integrate the function:

e^2x+3

Answer:

Let 2x + 3 = t 

\(\therefore \) 2dx = dt

39. Integrate the function:

\(\frac x{e^{x^2}}\)

Answer:

Let x² = t 

\(\therefore \) 2xdx = dt

40. Integrate the function:

\(\cfrac{e^{tan^{-1}}x}{1 +x^2}\)

Answer:

Let tan^-1 x = t

 \(\therefore \) \(\frac1{1 +x^2}\) dx = dt

41. Integrate the function:

\(\frac{e^{2x} - 1}{e^{2x } + 1}\)

Answer:

\(\frac{e^{2x} - 1}{e^{2x } + 1}\)

Dividing numerator and denominator by e^x , we obtain

42. Integrate the function:

\(\frac{e^{2x} - e^{-2x}}{e^{2x}+ e^{-2x}}\)

Answer:

Let e^2x + e^-2x = t

(2e^2x - 2e^-2x) dx =  dt

43. Integrate the function:

tan²(2x - 3)

Answer:

tan²(2x - 3) = sec²(2x - 3) - 1

Let 2x − 3 = t 

\(\therefore\) 2dx = dt

44. Integrate the function:

sec²(7 – 4x)

Answer:

Let 7 − 4x = t 

\(\therefore\) −4dx = dt

45. Integrate the function:

\(\frac{sin^{-1}x }{\sqrt{1 - x^2}}\)

Answer:

Let sin^-1 x = t

\(\frac{1 }{\sqrt{1 - x^2}}\)dx = dt

46. Integrate the function:

\(\frac{2\,cos\,x - 3\,sin\,x}{6\,cos\,x + 4\,sin\, x}\)

Answer:

47. Integrate the function:

\(\frac1{cos^2 x(1 - tan\,x )^2}\)

Answer:

48. Integrate the function:

\(\frac{cos\sqrt x}{\sqrt x}\)

Answer:

49. Integrate the function:

\(\sqrt{sin\,2x}\;cos\,2x\)

Answer:

Let sin 2x = t 

So, 2cos 2x dx = dt

50. Integrate the function:

\(\frac{cos\, x}{\sqrt{1 + sin\, x}}\)

Answer:

Let 1 + sin x = t 

\(\therefore\) cos x dx = dt

51. Integrate the function:

cot x log sin x

Answer:

Let log sin x = t

52. Integrate the function:

\(\frac{sin \,x}{1 + cos \,x}\)

Answer:

Let 1 + cos x = t 

\(\therefore\) −sin x dx = dt

Answer 4

53. Integrate the function:

\(\frac{sin\, x}{(1 + cos\, x)^2}\)

Answer:

Let 1 + cos x = t 

\(\therefore\) −sin x dx = dt

54. Integrate the function:

\(\frac1{1 + cot\,x}\)

Answer:

Let sin x + cos x = t 

⇒ (cos x − sin x) dx = dt

55. Integrate the function:

\(\frac1{1 - tan \, x}\)

Answer:

Put cos x − sin x = t 

⇒ (−sin x − cos x) dx = dt

56. Integrate the function:

\(\frac{\sqrt{tan\, x}}{sin\, x\;cos\, x}\)

Answer:

57. Integrate the function:

\(\frac{(1+ log\, x)^2}{x}\)

Answer:

Let 1 + log x = t

58. Integrate the function:

\(\frac{(x + 1) (x + log\,x)^2}{x}\)

Answer:

59. Integrate the function:

\(\frac{x^3 sin(tan^{-1}x^4)}{1 + x^8}\) 

Answer:

Let x⁴ = t 

\(\therefore\) 4x³ dx = dt

From (1), we obtain

60. \(\int \frac{10x^9 + 10^x log_e 10}{x^{10}10^x }dx\) equals 

(A) 10^x - x¹⁰ + C

(B) 10^x + x¹⁰ + C

(C) (10^x - x¹⁰)^-1 + C

(D) log (10^x + x¹⁰) + C

Answer:

Hence, the correct Answer is D.

61. \(\int \frac{dx}{sin^2x\,cos^2x}\) 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

Answer:

Hence, the correct Answer is B.

62. Find the integrals of the function:

sin²(2x + 5)

Answer:

63. Find the integrals of the function:

sin3x.cos4x

Answer:

It is known that,

64. Find the integrals of the function:

cos 2x cos 4x cos 6x

Answer:

It is known that,

65. Find the integrals of the function:

sin³(2x + 1)

Answer:

66. Find the integrals of the function:

sin³ x cos³ x

Answer:

67. Find the integrals of the function:

sin x sin 2x sin 3x

Answer:

It is known that,

68. Find the integrals of the function:

sin 4x sin 8x

Answer:

It is known that,

69. Find the integrals of the function:

\(\frac{1 - cos\,x}{1 + cos\,x}\)

Answer:

70. Find the integrals of the function:

\(\frac{cos\, x}{1 + cos\, x}\)

Answer:

71. Find the integrals of the function:

sin⁴x

Answer:

72. Find the integrals of the function:

cos⁴2x

Answer:

73. Find the integrals of the function:

\(\frac{sin^2x}{1 + cos x}\)

Answer:

Answer 5

74. Find the integrals of the function:

\(\frac{cos\,2x - cos\,2a}{cos\,x - cos\, a}\)

Answer:

75. Find the integrals of the function:

\(\frac{cos\, x - sin\,x}{1 + sin\,2x}\)

Answer:

76. Find the integrals of the function:

tan³2x sec 2x

Answer:

77. Find the integrals of the function:

tan⁴x

Answer:

From equation (1), we obtain

78. Find the integrals of the function:

\(\frac{sin^3 x+ cos ^3 x}{sin^2 x \,cos^2 x}\)

Answer:

79. Find the integrals of the function:

\(\frac{cos\,2x + 2sin^2 x}{cos^2x}\)

Answer:

80. Find the integrals of the function:

\(\frac{1}{sin\,x\; cos^3 x}\)

Answer:

81. Find the integrals of the function:

\(\frac{cos\, 2x}{(cos\,x + sin\, x)^2}\)

Answer:

82. Find the integrals of the function:

sin⁻¹ (cos x)

Answer:

It is known that,

Substituting in equation (1), we obtain

83. Find the integrals of the function:

\(\frac1{cos(x -a) cos(x - b)}\)

Answer:

84. \(\int \frac{sin^2x - cos^2 x}{sin^2 x \,cos^2x} dx\) is equal to 

(A) tan x + cot x + C 

(B) tan x + cosec x + C 

(C) - tan x + cot x + C 

(D) tan x + sec x + C

Answer:

Hence, the correct Answer is A.

85. \(\int\frac{e ^x(1 +x)}{cos^2 (e^x x)}dx\) equals

(A) – cot (ex^x) + C

(B) tan (xe^x) + C

(C) tan (e^x) + C

(D) cot (e^x) + C

Answer:

\(\int\frac{e ^x(1 +x)}{cos^2 (e^x x)}dx\)

Let ex^x = t

Hence, the correct Answer is B.

86. Integrate the function:

\(\frac{3x^2}{x^6 + 1}\)

Answer:

Let x³ = t 

∴ 3x² dx = dt

87. Integrate the function:

\(\frac1{\sqrt{1 + 4x^2}}\)

Answer:

Let 2x = t 

∴ 2dx = dt

88. Integrate the function:

\(\frac1{\sqrt{(2 - x)^2+1}}\)

Answer:

Let 2 − x = t 

⇒ −dx = dt

89. Integrate the function:

\(\frac{1}{\sqrt{9 - 25 x^2 }}\)

Answer:

Let 5x = t 

∴ 5dx = dt

90. Integrate the function:

\(\frac{3x}{1 + 2x^4}\)

Answer:

Let √2x² = t

∴ 2√2x dx = dt

91. Integrate the function:

\(\frac{x^2}{1 - x^6}\)

Answer:

Let x³ = t 

∴ 3x² dx = dt

92. Integrate the function:

\(\frac{x -1}{\sqrt{x^2 - 1}}\)

Answer:

From (1), we obtain

Answer 6

93. Integrate the function:

\(\frac{x^2 }{\sqrt{x^6 + a^6}}\)

Answer:

Let x³ = t 

⇒ 3x² dx = dt

94. Integrate the function:

\(\frac{sec^2 x}{\sqrt{tan^2 x + 4}}\)

Answer:

Let tan x = t 

∴ sec²x dx = dt

95. Integrate the function:

\(\frac{1}{\sqrt{x^2+ 2x + 2}}\)

Answer:

96. Integrate the function:

\(\frac1{\sqrt{9x^2 + 6x + 5}}\)

Answer:

97. Integrate the function:

\(\frac1{\sqrt{7 - 6x - x^2}}\)

Answer:

7 - 6x - x² can be written as 7 - (x² + 6x + 9 - 9)

Therefore,

98. Integrate the function:

\(\frac{1}{\sqrt{(x - 1) (x - 2)}}\)

Answer:

(x - 1) (x - 2) can be written as x² - 3x + 2.

Therefore,

99. Integrate the function:

\(\frac1{\sqrt{8 + 3x - x^2}}\)

Answer:

8 + 3x - x² can be written as \( 8 - \left(x^2 - 3x + \frac94 - \frac94\right)\)

Therefore,

100. Integrate the function:

\(\frac{1}{\sqrt{(x -a)(x - b)}}\)

Answer:

(x - a) (x - b) can be written as x² - (a + b)x + ab.

Therefore,

101. Integrate the function:

\(\frac{4x + 1}{\sqrt{2x^2 + x - 3}}\)

Answer:

Equating the coefficients of x and constant term on both sides, we obtain 

4A = 4 ⇒ A = 1

A + B = 1 ⇒ B = 0

Let 2x² + x − 3 = t

∴ (4x + 1) dx = dt

102. Integrate the function:

\(\frac{x + 2}{\sqrt{x^2 - 1}}\)

Answer:

Let x + 2 = A\(\frac d{dx}\)(x² - 1) + B     ....(1)

⇒ x + 2 = A (2x) + B

Equating the coefficients of x and constant term on both sides, we obtain

2A = 1

⇒ A = \(\frac12\)

B = 2

From (1), we obtain

From equation (2), we obtain

103. Integrate the function:

\(\frac{6x + 7}{\sqrt{(x - 5)(x - 4)}}\)

Answer:

Equating the coefficients of x and constant term, we obtain 

2A = 6 

⇒ A = 3 

−9A + B = 7 

⇒ B = 34 

∴ 6x + 7 = 3 (2x − 9) + 34

x² - 9x + 20 can be written as x² - 9x + 20 + \(\frac{81}4 - \frac{81}4\).

Therefore,

Substituting equations (2) and (3) in (1), we obtain

104. Integrate the function:

\(\frac{x + 2}{\sqrt{4x - x^2}}\)

Answer:

Let x + 2 = A\(\frac d{dx}\)(4x - x²) + B

⇒ x + 2 = A(4 - 2x) + B

Equating the coefficients of x and constant term on both sides, we obtain

Using equations (2) and (3) in (1), we obtain

105. Integrate the function:

\(\frac{x + 2}{\sqrt{x^2+ 2x + 3}}\)

Answer:

Let x² + 2x +3 = t 

⇒ (2x + 2) dx =dt

Using equations (2) and (3) in (1), we obtain

Answer 7

106. Integrate the function:

\(\frac{x + 3}{x^2 - 2x - 5}\) 

Answer:

Let (x + 3) = A\(\frac d{dx}\)(x² - 2x - 5) + B

(x + 3) = A(2x - 2) + B

Equating the coefficients of x and constant term on both sides, we obtain

Substituting (2) and (3) in (1), we obtain

107. Integrate the function:

\(\frac{5x + 3}{\sqrt{x^2 + 4x + 10}}\)

Answer:

Let 5x + 3 = A\(\frac d{dx}\)(x² + 4x + 10) + B

⇒ 5x + 3 = A(2x + 4) + B

Equating the coefficients of x and constant term, we obtain

Using equations (2) and (3) in (1), we obtain

108. \(\int \frac{dx}{x^2 + 2x + 2}\) equals

(A) x tan^–1 (x + 1) + C 

(B) tan^–1 (x + 1) + C 

(C) (x + 1) tan^–1x + C 

(D) tan^–1x + C

Answer:

Hence, the correct Answer is B.

109. \(\int \frac{dx}{\sqrt{9x - 4x^2}}\) equals

(A) \(\frac19 sin ^{-1}\left(\frac{9x - 8}{8}\right) +C\)

(B) \(\frac12 sin ^{-1}\left(\frac{8x - 9}{9}\right) +C\)

(C) \(\frac13 sin ^{-1}\left(\frac{9x - 8}{8}\right) +C\)

(D) \(\frac12 sin ^{-1}\left(\frac{9x - 8}{9}\right) +C\) 

Answer:

Hence, the correct Answer is B.

110. Integrate the rational function:

\(\frac x{(x +1)(x +2)}\)

Answer:

Let

Equating the coefficients of x and constant term, we obtain 

A + B = 1 

2A + B = 0 

On solving, we obtain 

A = −1 and B = 2

111. Integrate the rational function:

\(\frac1{x^2 -9}\)

Answer:

Let

Equating the coefficients of x and constant term, we obtain 

A + B = 0

−3A + 3B = 1 

On solving, we obtain

112. Integrate the rational function:

\(\frac{3x - 1}{(x - 1)(x -2)(x -3)}\)

Answer:

Let

Equating the coefficients of x² , x and constant term, we obtain 

A + B + C = 0 – 5A – 4B – 3C = 3 

6A + 3B + 2C = – 1 

Solving these equations, we obtain 

A = 1, B = −5, and C = 4

113. Integrate the rational function:

\(\frac x{(x - 1)(x - 2)(x - 3)}\)

Answer:

Let

Equating the coefficients of x², x and constant term, we obtain

A + B + C = 0

– 5A – 4B – 3C = 1

6A + 4B + 2C = 0

Solving these equations, we obtain

114. Integrate the rational function:

\(\frac{2x}{x^2 + 3x + 2}\)

Answer:

Let

Equating the coefficients of x², x and constant term, we obtain 

A + B = 2

2A + B = 0

Solving these equations, we obtain

A = −2 and B = 4

115. Integrate the rational function:

\(\frac{1 -x^2}{x(1 - 2x)}\)

Answer:

It can be seen that the given integrand is not a proper fraction. 

Therefore, on dividing (1 − x²) by x(1 − 2x), we obtain

Equating the coefficients of x², x and constant term, we obtain

– 2A + B = – 1 

And 

A = 2 

Solving these equations, we obtain 

A = 2 and B = 3

Substituting in equation (1), we obtain

Answer 8

 116. Integrate the rational function:

\(\frac x{(x^2 + 1) (x - 1)}\)

Answer:

Equating the coefficients of x², x, and constant term, we obtain 

A + C = 0 

−A + B = 1 

−B + C = 0 

On solving these equations, we obtain

\(A = \frac12, B= \frac12 \; and \; C = \frac12\)

From equation (1), we obtain

 117. Integrate the rational function:

\(\frac x{(x - 1)^2 (x + 2)}\)

Answer:

Substituting x = 1, we obtain 

Equating the coefficients of x², x and constant term, we obtain 

A + C = 0 

A + B – 2C = 1 

−2A + 2B + C = 0 

On solving, we obtain

 118. Integrate the rational function:

\(\frac{3x + 5}{x^3 - x^2 - x+ 1}\)

Answer:

Equating the coefficients of x² , x and constant term, we obtain 

A + C = 0 

B − 2C = 3 –

 A + B + C = 5 

On solving, we obtain B = 4

\(A = -\frac12 \; and\; C = \frac12\)

119. Integrate the rational function:

\(\frac{2x - 3}{(x^2 - 1)(2x + 3)}\)

Answer:

Equating the coefficients of x2 , x and constant, we obtain 

2A + 2B + C = 0 

A + 5B = 2 

– 3A + 3B – C = – 3

On solving, we obtain

\(B = -\frac1{10}, A = \frac52\;and\; C = -\frac{24}5\)

120. Integrate the rational function:

\(\frac{5x}{(x + 1) (x^2 - 4)}\)

Answer:

Equating the coefficients of x², x and constant, we obtain 

A + B + C = 0 

– B + 3C = 5 and 

– 4A – 2B + 2C = 0 

On solving, we obtain

121. Integrate the rational function:

\(\frac{x^3 + x + 1}{x^2 - 1}\)

Answer:

It can be seen that the given integrand is not a proper fraction. 

Therefore, on dividing (x³ + x + 1) by x² − 1, we obtain

Equating the coefficients of x and constant, we obtain 

A + B = 2 

– A + B = 1 

On solving, we obtain

 122. Integrate the rational function:

\(\frac{2}{(1 - x)(1 + x^2)}\)

Answer:

Equating the coefficient of x2 , x, and constant term, we obtain 

A − B = 0 

B − C = 0 

A + C = 2 

On solving these equations, we obtain 

A = 1, B = 1, and C = 1

123. Integrate the rational function:

\(\frac{3x - 1}{(x + 2)^2}\)

Answer:

Equating the coefficient of x and constant term, we obtain 

A = 3 

2A + B = −1 

⇒ B = −7

 124. Integrate the rational function:

\(\frac1{x^4 - 1}\)

Answer:

Equating the coefficient of x³, x², x, and constant term, we obtain

On solving these equations, we obtain

125. Integrate the rational function:

\(\frac1{x(x^n + 1)}\)

[Hint: multiply numerator and denominator by xⁿ⁻¹ and put xⁿ = t]

Answer:

\(\frac1{x(x^n + 1)}\)

Multiplying numerator and denominator by xⁿ⁻¹, we obtain

Equating the coefficients of t and constant, we obtain 

A = 1 and B = −1

126. Integrate the rational function:

\(\frac{cos\, x}{(1 -sin\,x)(2 - sin\, x)}\)

[Hint: Put sin x = t]

Answer:

Equating the coefficients of t and constant, we obtain 

– 2A – B = 0 and 2A + B = 1 

On solving, we obtain A = 1 and B = −1

Answer 9

127. Integrate the rational function:

\(\frac{(x^2 + 1)(x^2 + 2)}{(x^2 +3)(x^2 + 4)}\)

Answer:

Equating the coefficients of x³, x², x, and constant term, we obtain

A + C = 0 

B + D = 4 

4A + 3C = 0 

4B + 3D = 10 

On solving these equations, we obtain 

A = 0, B = −2, C = 0, and D = 6

128. Integrate the rational function:

\(\frac{2x}{(x^2 + 1)(x^2 + 3)}\)

Answer:

\(\frac{2x}{(x^2 + 1)(x^2 + 3)}\)

Let x² = t 

⇒ 2x dx = dt

Equating the coefficients of t and constant, we obtain 

A + B = 0 and 3A + B = 1 

On solving, we obtain

129. Integrate the rational function:

\(\frac1{x(x^4 - 1)}\)

Answer:

\(\frac1{x(x^4 - 1)}\)

Multiplying numerator and denominator by x³, we obtain

Let x⁴ = t 

⇒ 4x³dx = dt

Equating the coefficients of t and constant, we obtain 

A = −1 and B = 1

130. Integrate the rational function:

\(\frac1{(e^x -1)}\)

[Hint: Put e^x = t]

Answer:

\(\frac1{(e^x -1)}\)

Let ex = t 

⇒ e^x dx = dt

Equating the coefficients of t and constant, we obtain 

A = −1 and B = 1

131. \(\int \frac{xdx}{(x - 1)(x - 2)}\) equals

Answer:

Equating the coefficients of x and constant, we obtain 

A = −1 and B = 2

Hence, the correct Answer is B.

132. \(\int \frac{dx}{x(x^2 - 1)}\) equals

Answer:

Equating the coefficients of x², x, and constant term, we obtain 

A + B = 0 

C = 0 

A = 1

On solving these equations, we obtain 

A = 1, B = −1, and C = 0

Hence, the correct Answer is A.

133. Integrate the function:

x sin x

Answer:

Let I = \(\int x \,sin \,x \;dx\)

Taking x as first function and sin x as second function and integrating by parts, we obtain

134. Integrate the function:

x sin 3x

Answer:

Let I = \(\int x \,sin \,3x \;dx\)

Taking x as first function and sin 3x as second function and integrating by parts, we obtain

135. Integrate the function:

x² e^x

Answer:

Let I = \(\int x^2 e^x \;dx\)

Taking x² as first function and e^x as second function and integrating by parts, we obtain

Again integrating by parts, we obtain

136. Integrate the function:

x logx

Answer:

Let I = \(\int x\,log\,x\;dx\)

Taking log x as first function and x as second function and integrating by parts, we obtain

137. Integrate the function:

x log 2x

Answer:

Let I = \(\int x\,log\,2x\;dx\)

Taking log 2x as first function and x as second function and integrating by parts, we obtain

138. Integrate the function:

x² log x

Answer:

Let I = \(\int x^2\,log\,x\;dx\)

Taking log x as first function and x² as second function and integrating by parts, we obtain

Answer 10

139. Integrate the function:

x sin^-1x

Answer:

Let I = \(\int x\, sin^{-1}x\;dx\) 

Taking sin^-1x as first function and x as second function and integrating by parts, we obtain

140. Integrate the function:

x tan^-1x

Answer:

Let I = \(\int x\, tan^{-1}x\;dx\)

Taking tan^-1x as first function and x as second function and integrating by parts, we obtain

141. Integrate the function:

x cos^-1 x

Answer:

Let I = \(\int x\, cos^{-1}x\;dx\)

Taking cos⁻¹ x as first function and x as second function and integrating by parts, we obtain

Substituting in (1), we obtain

142. Integrate the function:

(sin^-1x)²

Answer:

Let I = \(\int (sin^{-1}x)^2 .1 \;dx\)

Taking ( sin–1x)² as first function and 1 as second function and integrating by parts, we obtain

143. Integrate the function:

\(\frac{x\, cos^{-1}x}{\sqrt{1 -x^2}}\)

Answer:

Let I = \(\int \frac{x\, cos^{-1}x}{\sqrt{1 -x^2}}dx\)

\( I = \frac{-1}2\int \frac{-2x}{\sqrt{1 -x^2}}.cos^{-1}x\;dx\)

Taking cos^-1x as first function and \(\left(\frac{-2x}{\sqrt{1 - x^2}}\right)\) as second function and integrating by parts, we obtain

144. Integrate the function:

x sec²x

Answer:

Let I = \(\int x \, sec^2\; dx\)

Taking x as first function and sec²x as second function and integrating by parts, we obtain

145. Integrate the function:

tan^–1x

Answer:

Let I = \(\int 1. tan^{-1}x\;dx\)

Taking tan^–1x as first function and 1 as second function and integrating by parts, we obtain

146. Integrate the function:

x (log x)²

Answer:

Taking (log x)² as first function and 1 as second function and integrating by parts, we obtain

Again integrating by parts, we obtain

147. Integrate the function:

(x² + 1) log x

Answer:

Taking log x as first function and x2 as second function and integrating by parts, we obtain

Taking log x as first function and 1 as second function and integrating by parts, we obtain

Using equations (2) and (3) in (1), we obtain

148. Integrate the function:

e^x(sin x + cos x)

Answer:

Let \(I = \int e^x (sin x + cos x)dx\)

Let \(f(x) = sin x\)

\(f'(x) = cos x\)

\( I = \int e^x [f(x) + f'(x)]dx\)

It is known that,

\( \int e^x [f(x) + f'(x)]dx= e^x f(x) + C\)

\(\therefore I = e^x sin\, x + C\) 

149. Integrate the function:

\(e^x \left(\frac{1 + sin\, x}{1 + cos\,x}\right)\)

Answer:

Let \(tan \frac x2 = f(x)\) So \(f'(x) = \frac12 sec^2\frac x2\)

It is known that, \(\int e^x [f(x) + f'(x)] dx = e^xf(x) + C\)

From equation (1), we obtain 

150. Integrate the function:

\(e^x \left(\frac 1x - \frac1{x^2}\right)\)

Answer:

Let I = \(\int e^x \left(\frac 1x - \frac1{x^2}\right)dx\) 

Also, let \(\frac 1x = f(x)\) \(f'(x) = \frac {-1}{x^2}\)

It is known that,

\(\int e^x [f(x) + f'(x)] dx = e^x f(x) + C\)

\(\therefore I = \frac{e^x}{x} + C\)

151. Integrate the function:

\(\frac{(x - 3)e^x}{(x - 1)^3}\)

Answer:

152. Integrate the function:

e^2x sin x

Answer:

Let \(I = \int e^{2x} sin \, x \)     .....(1)

Integrating by parts, we obtain

Again integrating by parts, we obtain