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NCERT Solutions › Mathematics › Chapter 7

Chapter 7: Integrals

NCERT Solutions for CBSE Class 10 Mathematics — 81 solved questions with detailed explanations.

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Solved Questions

Q1. ∫ x² dx equals:

Difficulty: Easy · Topic: Basic Integration

Solution

∫ xⁿ dx = x^(n+1)/(n+1) + C. For n=2: x³/3 + C.

Q2. ∫₀^π sin x dx equals:

Difficulty: Easy · Topic: Definite Integral

Solution

[-cos x]₀^π = -cosπ + cos0 = 1 + 1 = 2.

Q3. ∫ sin x dx =

Difficulty: Easy · Topic: Standard Integrals

Solution

- cos x + C.

Q4. ∫ cos x dx =

Difficulty: Easy · Topic: Standard Integrals

Solution

sin x + C.

Q5. ∫ sec²x dx =

Difficulty: Easy · Topic: Standard Integrals

Solution

tan x + C.

Q6. ∫ cosec²x dx =

Difficulty: Easy · Topic: Standard Integrals

Solution

-cot x + C.

Q7. ∫ e^x dx =

Difficulty: Easy · Topic: Standard Integrals

Solution

e^x + C.

Q8. ∫ 1/x dx =

Difficulty: Easy · Topic: Standard Integrals

Solution

ln|x| + C.

Q9. ∫ sec x tan x dx =

Difficulty: Easy · Topic: Standard Integrals

Solution

sec x + C.

Q10. ∫ from 0 to 1 of x² dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

Computing the definite integral gives 1/3.

Q11. ∫ from 0 to 2 of x³ dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

Computing the definite integral gives 4.

Q12. ∫ from 1 to 3 of x dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

Computing the definite integral gives 4.

Q13. ∫ from 0 to 1 of x dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

Computing the definite integral gives 0.5.

Q14. ∫ from 0 to 4 of x½ dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

Computing the definite integral gives 16/3.

Q15. ∫ from 0 to 1 of x^1 dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

[x^2/2] from 0 to 1 = 1^2/2 = 0.5.

Q16. ∫ from 0 to 2 of x^1 dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

[x^2/2] from 0 to 2 = 2^2/2 = 2.0.

Q17. ∫ from 0 to 3 of x^1 dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

[x^2/2] from 0 to 3 = 3^2/2 = 4.5.

Q18. ∫ from 0 to 1 of x^2 dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

[x^3/3] from 0 to 1 = 1^3/3 = 0.3333.

Q19. ∫ from 0 to 2 of x^2 dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

[x^3/3] from 0 to 2 = 2^3/3 = 2.6667.

Q20. ∫ from 0 to 3 of x^2 dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

[x^3/3] from 0 to 3 = 3^3/3 = 9.0.

Q21. ∫ from 0 to 1 of x^3 dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

[x^4/4] from 0 to 1 = 1^4/4 = 0.25.

Q22. ∫ from 0 to 2 of x^3 dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

[x^4/4] from 0 to 2 = 2^4/4 = 4.0.

Q23. ∫ from 0 to 3 of x^3 dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

[x^4/4] from 0 to 3 = 3^4/4 = 20.25.

Q24. ∫ from 0 to 1 of x^4 dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

[x^5/5] from 0 to 1 = 1^5/5 = 0.2.

Q25. ∫ from 0 to 2 of x^4 dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

[x^5/5] from 0 to 2 = 2^5/5 = 6.4.

Q26. ∫ from 0 to 3 of x^4 dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

[x^5/5] from 0 to 3 = 3^5/5 = 48.6.

Q27. ∫ from 0 to 1 of x^5 dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

[x^6/6] from 0 to 1 = 1^6/6 = 0.1667.

Q28. ∫ from 0 to 2 of x^5 dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

[x^6/6] from 0 to 2 = 2^6/6 = 10.6667.

Q29. ∫ from 0 to 3 of x^5 dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

[x^6/6] from 0 to 3 = 3^6/6 = 121.5.

Q30. ∫ from 0 to 1 of x^6 dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

[x^7/7] from 0 to 1 = 1^7/7 = 0.1429.

Q31. ∫ from 0 to 2 of x^6 dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

[x^7/7] from 0 to 2 = 2^7/7 = 18.2857.

Q32. ∫ from 0 to 3 of x^6 dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

[x^7/7] from 0 to 3 = 3^7/7 = 312.4286.

Q33. ∫ from 0 to 1 of x^7 dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

[x^8/8] from 0 to 1 = 1^8/8 = 0.125.

Q34. ∫ from 0 to 2 of x^7 dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

[x^8/8] from 0 to 2 = 2^8/8 = 32.0.

Q35. ∫ from 0 to 3 of x^7 dx =

Difficulty: Easy-Medium · Topic: Definite Integrals

Solution

[x^8/8] from 0 to 3 = 3^8/8 = 820.125.

Q36. ∫(0 to 1) x^1 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^2/2]_0^1 = 1^2/2 = 0.5.

Q37. ∫(0 to 2) x^1 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^2/2]_0^2 = 2^2/2 = 2.0.

Q38. ∫(0 to 3) x^1 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^2/2]_0^3 = 3^2/2 = 4.5.

Q39. ∫(0 to 4) x^1 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^2/2]_0^4 = 4^2/2 = 8.0.

Q40. ∫(0 to 5) x^1 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^2/2]_0^5 = 5^2/2 = 12.5.

Q41. ∫(0 to 1) x^2 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^3/3]_0^1 = 1^3/3 = 0.3333.

Q42. ∫(0 to 2) x^2 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^3/3]_0^2 = 2^3/3 = 2.6667.

Q43. ∫(0 to 3) x^2 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^3/3]_0^3 = 3^3/3 = 9.0.

Q44. ∫(0 to 4) x^2 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^3/3]_0^4 = 4^3/3 = 21.3333.

Q45. ∫(0 to 5) x^2 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^3/3]_0^5 = 5^3/3 = 41.6667.

Q46. ∫(0 to 1) x^3 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^4/4]_0^1 = 1^4/4 = 0.25.

Q47. ∫(0 to 2) x^3 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^4/4]_0^2 = 2^4/4 = 4.0.

Q48. ∫(0 to 3) x^3 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^4/4]_0^3 = 3^4/4 = 20.25.

Q49. ∫(0 to 4) x^3 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^4/4]_0^4 = 4^4/4 = 64.0.

Q50. ∫(0 to 5) x^3 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^4/4]_0^5 = 5^4/4 = 156.25.

Q51. ∫(0 to 1) x^4 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^5/5]_0^1 = 1^5/5 = 0.2.

Q52. ∫(0 to 2) x^4 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^5/5]_0^2 = 2^5/5 = 6.4.

Q53. ∫(0 to 3) x^4 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^5/5]_0^3 = 3^5/5 = 48.6.

Q54. ∫(0 to 4) x^4 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^5/5]_0^4 = 4^5/5 = 204.8.

Q55. ∫(0 to 5) x^4 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^5/5]_0^5 = 5^5/5 = 625.0.

Q56. ∫(0 to 1) x^5 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^6/6]_0^1 = 1^6/6 = 0.1667.

Q57. ∫(0 to 2) x^5 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^6/6]_0^2 = 2^6/6 = 10.6667.

Q58. ∫(0 to 3) x^5 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^6/6]_0^3 = 3^6/6 = 121.5.

Q59. ∫(0 to 4) x^5 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^6/6]_0^4 = 4^6/6 = 682.6667.

Q60. ∫(0 to 5) x^5 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^6/6]_0^5 = 5^6/6 = 2604.1667.

Q61. ∫(0 to 1) x^6 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^7/7]_0^1 = 1^7/7 = 0.1429.

Q62. ∫(0 to 2) x^6 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^7/7]_0^2 = 2^7/7 = 18.2857.

Q63. ∫(0 to 3) x^6 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^7/7]_0^3 = 3^7/7 = 312.4286.

Q64. ∫(0 to 4) x^6 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^7/7]_0^4 = 4^7/7 = 2340.5714.

Q65. ∫(0 to 5) x^6 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^7/7]_0^5 = 5^7/7 = 11160.7143.

Q66. ∫(0 to 1) x^7 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^8/8]_0^1 = 1^8/8 = 0.125.

Q67. ∫(0 to 2) x^7 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^8/8]_0^2 = 2^8/8 = 32.0.

Q68. ∫(0 to 3) x^7 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^8/8]_0^3 = 3^8/8 = 820.125.

Q69. ∫(0 to 4) x^7 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^8/8]_0^4 = 4^8/8 = 8192.0.

Q70. ∫(0 to 5) x^7 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^8/8]_0^5 = 5^8/8 = 48828.125.

Q71. ∫(0 to 1) x^8 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^9/9]_0^1 = 1^9/9 = 0.1111.

Q72. ∫(0 to 2) x^8 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^9/9]_0^2 = 2^9/9 = 56.8889.

Q73. ∫(0 to 3) x^8 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^9/9]_0^3 = 3^9/9 = 2187.0.

Q74. ∫(0 to 4) x^8 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^9/9]_0^4 = 4^9/9 = 29127.1111.

Q75. ∫(0 to 5) x^8 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^9/9]_0^5 = 5^9/9 = 217013.8889.

Q76. ∫(0 to 1) x^9 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^10/10]_0^1 = 1^10/10 = 0.1.

Q77. ∫(0 to 2) x^9 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^10/10]_0^2 = 2^10/10 = 102.4.

Q78. ∫(0 to 3) x^9 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^10/10]_0^3 = 3^10/10 = 5904.9.

Q79. ∫(0 to 4) x^9 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^10/10]_0^4 = 4^10/10 = 104857.6.

Q80. ∫(0 to 5) x^9 dx =

Difficulty: Easy-Medium · Topic: Definite

Solution

[x^10/10]_0^5 = 5^10/10 = 976562.5.

Q81. ∫ e^x(sin x + cos x) dx equals:

Difficulty: Medium · Topic: Integration by Parts

Solution

Using ∫ e^x[f(x)+f'(x)]dx = e^x f(x) + C. Here f(x) = sinx, f'(x) = cosx.

Other Chapters in Mathematics

Ch 1: Relations & FunctionsCh 2: Inverse Trig FunctionsCh 3: MatricesCh 4: DeterminantsCh 5: Continuity & DifferentiabilityCh 6: Applications of DerivativesCh 8: Applications of IntegralsCh 9: Differential EquationsCh 10: Vector AlgebraCh 11: Three Dimensional GeometryCh 12: Linear ProgrammingCh 13: Probability

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