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7

Q1. Integrate:

Solution

Q2. Evaluate:

Solution

Q3. Find the integral of the function  w. r. to x.

Solution

Q4. Integrate: .

Solution

Q5. Integrate .

Solution

Q6. The value of  is:
  • 1)
  • 2)
  • 3)
  • 4)

Solution

Q7. The value of is:
  • 1) 1
  • 2) 0
  • 3) 2
  • 4) -1

Solution

Q8. Find the integral:

Solution

Q9. Evaluate:

Solution

Q10. Integrate:

Solution

Q11. Evaluate:

Solution

Q12. Evaluate:

Solution

Q13. Evaluate:

Solution

Q14. Evaluate:

Solution

Q15. Find the integral of the function  w. r. to x.

Solution

Q16. Find the integral of the function  w. r. to x.

Solution

Q17. Integrate:

Solution

Q18. Evaluate:

Solution

Q19. Evaluate:

Solution

Q20. Evaluate:

Solution

Q21. Integrate: .

Solution

Q22. Find the integral of the function  w. r. to x.

Solution

Q23. Evaluate:

Solution

Q24. Evaluate:

Solution

Q25. Evaluate:

Solution

Q26. Integrate:  .

Solution

Q27. Evaluate: integral subscript 0 superscript straight pi over 2 end superscript log left parenthesis cos x right parenthesis d x.

Solution

L e blank over blank t space I equals integral subscript 0 superscript straight pi over 2 end superscript log left parenthesis cos x right parenthesis d x rightwards arrow circle enclose 1 space space space space space space space space equals integral subscript 0 superscript straight pi over 2 end superscript log left parenthesis cos open parentheses straight pi over 2 minus x close parentheses right parenthesis d x space space space space space space space space I equals integral subscript 0 superscript straight pi over 2 end superscript log left parenthesis sin x right parenthesis d x rightwards arrow circle enclose 2 a d d i n g space circle enclose 1 space a n d space circle enclose 2 space w e space g e t 2 I equals integral subscript 0 superscript straight pi over 2 end superscript left square bracket log left parenthesis sin x right parenthesis plus log left parenthesis cos x right parenthesis right square bracket d x space space space space equals integral subscript 0 superscript straight pi over 2 end superscript log open parentheses fraction numerator sin 2 x over denominator 2 end fraction close parentheses d x space space space space equals integral subscript 0 superscript straight pi over 2 end superscript open parentheses log sin 2 x minus log 2 close parentheses d x space space space space equals integral subscript 0 superscript straight pi over 2 end superscript log sin 2 x d x minus integral subscript 0 superscript straight pi over 2 end superscript log 2 d x space space space space space equals J minus straight pi over 2 log 2 w h e r e space J equals integral subscript 0 superscript straight pi over 2 end superscript logsin 2 x d x l e t space 2 x equals t rightwards double arrow d x equals d t divided by 2 x equals 0 comma t equals 0 x equals straight pi over 2 comma t equals straight pi straight J equals 1 half integral subscript 0 superscript straight pi log left parenthesis sint right parenthesis dt space space space using space the space property space integral subscript 0 superscript 2 straight a end superscript straight f left parenthesis straight x right parenthesis d straight x equals 2 integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis d straight x space if space straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 2 straight a minus straight x right parenthesis space we space can space write straight J equals 1 half 2 integral subscript 0 superscript straight pi over 2 end superscript log left parenthesis sint right parenthesis dt equals integral subscript 0 superscript straight pi over 2 end superscript log left parenthesis sint right parenthesis dt equals straight I 2 straight I equals straight J minus straight pi over 2 log 2 space space space space equals straight I minus straight pi over 2 log 2 straight I equals negative straight pi over 2 log 2    
Q28. Integrate: .

Solution

Q29. , find the value of k

Solution

Q30. Integrate:

Solution

Q31. Evaluate:

Solution

Q32. Integrate:

Solution

Q33. Integrate:

Solution

Q34. Evaluate:

Solution

Q35. Integrate:

Solution

Q36. Evaluate:

Solution

Q37. Evaluate:

Solution

Q38. Integrate: integral fraction numerator 1 over denominator sin squared x left parenthesis c o t x minus 1 right parenthesis squared left parenthesis c o t x plus 1 right parenthesis end fraction d x.

Solution

Q39. Integrate:

Solution

Q40. Integrate:

Solution

Q41. Integrate:

Solution

Q42. Integrate: .

Solution

Q43. Integrate: .

Solution

Q44. Integrate .

Solution

Q45. Evaluate:

Solution

Q46. Find the integral of the function w. r. to x.

Solution

Q47. Evaluate:

Solution

Q48. Integrate:

Solution

Q49. Find the integral of the function  w. r. to x

Solution

Q50. Integrate: .

Solution

Q51. Find the integral of the function w. r. to x.

Solution

Q52. Integrate .

Solution

Q53. Integrate:

Solution

Q54. Find the integral of the function  w. r. to x

Solution

Q55. Integrate:

Solution

Q56. Evaluate  as the limit of a sum

Solution

Here, a = – 1, b = 2 and
Q57. Integrate:

Solution

Q58. Integrate:

Solution



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