Integrals

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Question 1

Let space straight I space space equals space integral subscript 0 superscript 1 space fraction numerator sin space straight x over denominator square root of straight x end fraction dx space and space straight J space equals space integral subscript 0 superscript 1 space fraction numerator cos space straight x over denominator square root of straight x end fraction space dx then which one is true?
  • I > 2/ 3 and J > 2

  • I < 2/ 3 and J < 2

  • I < 2/ 3 and J > 2

  • I > 2/ 3 and J < 2

Solution

B.

I < 2/ 3 and J < 2

straight I equals space integral subscript 0 superscript 1 fraction numerator sin space straight x over denominator square root of straight x end fraction space dx space less than space integral subscript 0 superscript 1 space fraction numerator straight x over denominator square root of straight x end fraction dx space equals space integral subscript 0 superscript 1 square root of straight x dx space equals space 2 over 3 straight x to the power of 3 divided by 2 end exponent space left enclose space subscript 0 superscript 1 end enclose space equals space 2 over 3
rightwards double arrow straight l space less than space 2 over 3
straight J space equals space integral subscript 0 superscript 1 fraction numerator cos space straight x over denominator square root of straight x end fraction space dx space less than integral subscript 0 superscript 1 fraction numerator 1 over denominator square root of straight x end fraction space dx space equals space 2 square root of straight x vertical line subscript 0 superscript 1 space equals space 2
therefore space straight J less or equal than 2
Question 2

let space straight F left parenthesis straight x right parenthesis space equals space straight f left parenthesis straight x right parenthesis space plus space straight f space open parentheses 1 over straight x close parentheses comma space where space straight f left parenthesis straight x right parenthesis space equals space integral subscript 1 superscript straight x fraction numerator log space straight t over denominator 1 plus straight t end fraction dt. Then F(e) equals
  • 1/2

  • 1

  • 2

  • 0

Solution

A.

1/2

straight f left parenthesis straight x right parenthesis space equals space integral subscript 1 superscript straight x fraction numerator log space straight t over denominator 1 space plus straight t end fraction space dt
straight F left parenthesis straight e right parenthesis space equals space straight f left parenthesis straight e right parenthesis space plus space straight f open parentheses 1 over straight e close parentheses
straight F left parenthesis straight e right parenthesis space equals space integral subscript 1 superscript straight e space fraction numerator log space straight t over denominator 1 plus straight t end fraction dt space plus integral subscript 1 superscript 1 divided by straight e end superscript fraction numerator log space straight t over denominator 1 plus straight t end fraction dt
space equals space integral subscript 1 superscript straight e fraction numerator log space straight t over denominator 1 plus straight t end fraction space plus integral subscript 1 superscript straight e fraction numerator log space straight t over denominator straight t left parenthesis 1 plus straight t right parenthesis end fraction dt
space equals space integral subscript 1 superscript straight e space fraction numerator log space straight t over denominator straight t end fraction space dt space equals space 1 half
Question 3

integral subscript 0 superscript straight pi space xf space left parenthesis sin space straight x space right parenthesis space dx is equal to
  • straight pi integral subscript 0 superscript straight pi space straight f left parenthesis cos space straight x right parenthesis space dx
  • straight pi integral subscript 0 superscript straight pi space straight f left parenthesis sin space straight x right parenthesis space dx
  • straight pi over 2 integral subscript 0 superscript straight pi space straight f left parenthesis cos space straight x right parenthesis space dx
  • straight pi integral subscript 0 superscript straight pi divided by 2 end superscript space straight f left parenthesis cos space straight x right parenthesis space dx

Solution

D.

straight pi integral subscript 0 superscript straight pi divided by 2 end superscript space straight f left parenthesis cos space straight x right parenthesis space dx straight I space equals space integral subscript 0 superscript straight pi xf space left parenthesis sin space straight x right parenthesis space dx
equals space integral subscript 0 superscript straight pi left parenthesis straight pi minus straight x right parenthesis space straight f left parenthesis sin space straight x right parenthesis space dx
space equals integral subscript 0 superscript straight pi straight f left parenthesis sin space straight x right parenthesis space dx space minus straight l
2 straight l space equals space integral subscript 0 superscript straight pi straight f left parenthesis sin space straight x right parenthesis space dx
straight l space equals space straight pi over 2 integral subscript 0 superscript straight pi space straight f left parenthesis sin space straight x right parenthesis space dx
space equals straight pi space integral subscript 0 superscript straight pi divided by 2 end superscript space straight f space left parenthesis sin space straight x right parenthesis space dx
equals straight pi integral subscript 0 superscript straight pi divided by 2 end superscript space straight f left parenthesis cos space straight x right parenthesis space dx
Question 4

integral subscript negative 3 straight pi divided by 2 end subscript superscript negative straight pi divided by 2 end superscript open square brackets open parentheses straight x plus straight pi close parentheses cubed plus space cos space left parenthesis straight x space plus 3 straight pi right parenthesis close square brackets dx space is space equal space to
  • straight pi to the power of 4 over 32
  • straight pi to the power of 4 over 32 space plus space pi over 2
  • pi over 2
  • pi over 4 minus 1

Solution

C.

pi over 2 straight I space equals space integral subscript negative 3 straight pi divided by 2 end subscript superscript straight pi divided by 2 end superscript open square brackets open parentheses straight x plus straight pi close parentheses cubed space plus space cos squared space left parenthesis straight x space plus 3 straight pi right parenthesis close square brackets dx
space Put space straight x space plus straight pi space equals space straight t
straight I space equals space integral subscript negative straight pi divided by 2 end subscript superscript straight pi divided by 2 end superscript space open square brackets straight t cubed space plus space cos squared straight t close square brackets dt
space equals space 2 integral subscript 0 superscript straight pi divided by 2 end superscript space cos squared straight t space dt
space equals space integral subscript 0 superscript straight pi divided by 2 end superscript space left parenthesis 1 plus space cos space 2 straight t right parenthesis dt space equals space straight pi over 2 plus 0
Question 5

integral space open curly brackets fraction numerator left parenthesis log space straight x minus 1 right parenthesis over denominator 1 plus left parenthesis log space straight x right parenthesis squared end fraction close curly brackets squared space dx space is space equal space to
  • fraction numerator log space straight x over denominator left parenthesis log space straight x right parenthesis squared plus 1 end fraction plus space straight C
  • fraction numerator straight x over denominator straight x plus 1 end fraction space plus straight C
  • fraction numerator xe to the power of straight x over denominator 1 plus straight x squared end fraction plus straight C
  • fraction numerator straight x over denominator left parenthesis log space straight x right parenthesis squared space plus space 1 end fraction plus space straight C

Solution

D.

fraction numerator straight x over denominator left parenthesis log space straight x right parenthesis squared space plus space 1 end fraction plus space straight C integral fraction numerator left parenthesis log space straight x minus 1 right parenthesis squared over denominator left parenthesis 1 plus left parenthesis log space straight x right parenthesis squared right parenthesis squared end fraction dx
space equals space integral space open square brackets fraction numerator 1 over denominator left parenthesis 1 plus left parenthesis log space straight x right parenthesis squared right parenthesis end fraction minus fraction numerator 2 space log space straight x over denominator left parenthesis 1 plus left parenthesis log space straight x right parenthesis squared right parenthesis squared end fraction close square brackets space dx
space equals integral open square brackets fraction numerator straight e to the power of straight t over denominator 1 plus straight t squared end fraction minus fraction numerator 2 straight t space straight e to the power of straight t over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction close square brackets space dt space put space log space straight x space equals space straight t
rightwards double arrow space dx space equals straight e to the power of straight t space dt
integral straight e to the power of straight t space open square brackets fraction numerator 1 over denominator 1 plus straight t squared end fraction minus fraction numerator 2 straight t over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction close square brackets dt
space equals space fraction numerator straight e to the power of straight t over denominator 1 plus straight t squared end fraction space plus straight c equals space fraction numerator straight x over denominator 1 plus space left parenthesis log space straight x right parenthesis squared end fraction space plus straight c