is equal to from Mathematics Integrals

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21. 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 2
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22. 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 xe to the power of straight x over denominator 1 plus straight x squared end fraction plus straight C
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23.

Let f : R → R be a differentiable function having f (2) = 6, f′ (2) =(1/48) . Then limit as straight x space rightwards arrow 2 of space integral subscript 6 superscript straight f left parenthesis straight x right parenthesis end superscript space fraction numerator 4 straight t cubed over denominator straight x minus 2 end fraction dt space equals

  • 24

  • 36

  • 12

  • 12

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24.

Let f (x) be a non−negative continuous function such that the area bounded by the curve y = f (x), x−axis and the ordinates x = π/4  and x = β >  π/4 open parentheses straight beta space sin space straight beta space plus space straight pi over 4 space cos space straight beta space plus space square root of 2 straight beta close parentheses. Then f (π/2) is

  • open parentheses straight pi over 4 plus square root of 2 minus 1 close parentheses
  • open parentheses straight pi over 4 minus square root of 2 plus 1 close parentheses
  • open parentheses 1 minus straight pi over 4 minus square root of 2 plus 1 close parentheses
  • open parentheses 1 minus straight pi over 4 minus square root of 2 plus 1 close parentheses
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25.

The value of integral subscript negative straight pi end subscript superscript straight pi space fraction numerator cos squared space straight x over denominator 1 plus space straight a to the power of straight x end fraction space dx comma space straight a greater than 0 space is

  • a π

  • π/2

  • π/a

  • π/a

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26.

If integral space fraction numerator sin space straight x over denominator sin space left parenthesis straight x minus straight alpha right parenthesis end fraction space dx space equals space Ax space plus space straight B space log space sin space left parenthesis straight x minus straight alpha right parenthesis space plus straight C comma space space then space value space left parenthesis straight A comma space straight B right parenthesis is

  • (sinα, cosα)

  • (cosα, sinα)

  • (- sinα, cosα)

  • (- sinα, cosα)

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27. integral fraction numerator dx over denominator cos space straight x space minus space sin space straight x end fraction is equal to
  • fraction numerator 1 over denominator square root of 2 end fraction space log space open vertical bar tan open parentheses straight x over 2 minus fraction numerator begin display style straight pi end style over denominator 8 end fraction close parentheses close vertical bar space plus straight C
  • fraction numerator 1 over denominator square root of 2 end fraction space log space open vertical bar cot open parentheses straight x over 2 close parentheses close vertical bar space plus straight C
  • fraction numerator 1 over denominator square root of 2 end fraction space log space open vertical bar tan open parentheses straight x over 2 minus fraction numerator 3 straight pi over denominator 8 end fraction close parentheses close vertical bar space plus straight C
  • fraction numerator 1 over denominator square root of 2 end fraction space log space open vertical bar tan open parentheses straight x over 2 minus fraction numerator 3 straight pi over denominator 8 end fraction close parentheses close vertical bar space plus straight C


D.

fraction numerator 1 over denominator square root of 2 end fraction space log space open vertical bar tan open parentheses straight x over 2 minus fraction numerator 3 straight pi over denominator 8 end fraction close parentheses close vertical bar space plus straight C
integral fraction numerator dx over denominator cos space straight x space minus space sin space straight x end fraction space equals space fraction numerator 1 over denominator square root of 2 end fraction integral fraction numerator 1 over denominator cos begin display style space end style begin display style open parentheses straight x space plus straight pi over 4 close parentheses end style end fraction space dx

equals space fraction numerator 1 over denominator square root of 2 end fraction integral space sec space open parentheses space straight x plus straight pi over 4 space close parentheses dx
space equals space fraction numerator 1 over denominator square root of 2 end fraction space log space open vertical bar space tan space open parentheses straight x over 2 space plus fraction numerator 3 straight pi over denominator 8 end fraction close parentheses close vertical bar space plus straight C
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28.

The value of integral subscript negative 2 end subscript superscript 3 space vertical line 1 minus straight x squared vertical line dx space is

  • 28/7

  • 28/3

  • 7/3

  • 7/3

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29.

The value of integral subscript 0 superscript pi divided by 2 end superscript fraction numerator left parenthesis sin space x space plus space cos space x right parenthesis squared over denominator square root of 1 plus space sin space 2 x end root end fraction space d x space i s

  • 0

  • 1

  • 2

  • 2

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30.

Ifintegral subscript 0 superscript straight pi space xf space left parenthesis sin right parenthesis space dx space equals space straight A 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 comma space then space straight A space is

  • 0

  • π

  • π/2

  • π/2

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