The integral  is equal to  from Mathematics Integrals

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 Multiple Choice QuestionsMultiple Choice Questions

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1.

The integral integral fraction numerator 2 straight x to the power of 12 space plus space 5 straight x to the power of 9 over denominator left parenthesis straight x to the power of 5 space plus space straight x cubed space plus space 1 right parenthesis cubed end fraction space dx is equal to 

  • fraction numerator negative straight x to the power of 5 over denominator left parenthesis straight x to the power of 5 space plus straight x cubed plus 1 right parenthesis squared end fraction space plus straight C
  • fraction numerator straight x to the power of 10 over denominator 2 left parenthesis straight x to the power of 5 space plus straight x cubed plus 1 right parenthesis squared end fraction space plus straight C
  • fraction numerator straight x to the power of 5 over denominator 2 left parenthesis straight x to the power of 5 space plus straight x cubed plus 1 right parenthesis squared end fraction space plus straight C
  • fraction numerator straight x to the power of 5 over denominator 2 left parenthesis straight x to the power of 5 space plus straight x cubed plus 1 right parenthesis squared end fraction space plus straight C


B.

fraction numerator straight x to the power of 10 over denominator 2 left parenthesis straight x to the power of 5 space plus straight x cubed plus 1 right parenthesis squared end fraction space plus straight C
Let space straight I space space equals space integral fraction numerator 2 straight x to the power of 12 space plus 5 straight x to the power of 9 over denominator left parenthesis straight x to the power of 5 space plus straight x cubed space plus 1 right parenthesis end fraction dx
space equals space integral fraction numerator 2 straight x to the power of 12 space plus space 5 straight x to the power of 9 over denominator straight x to the power of 15 left parenthesis 1 space plus space straight x to the power of negative 2 end exponent space plus straight x to the power of negative 5 end exponent right parenthesis cubed end fraction space dx
space equals space integral space fraction numerator 2 straight x to the power of negative 3 end exponent space plus 5 straight x to the power of negative 6 end exponent over denominator left parenthesis 1 plus straight x to the power of negative 2 end exponent plus straight x to the power of negative 5 end exponent right parenthesis cubed end fraction dx
Now comma space put space 1 plus space straight x to the power of negative 2 end exponent space plus straight x to the power of negative 5 end exponent space equals space straight t
rightwards double arrow space left parenthesis negative 2 straight x to the power of negative 3 end exponent space minus 5 straight x to the power of negative 6 end exponent right parenthesis space dx space equals space dt
rightwards double arrow space left parenthesis 2 straight x to the power of negative 3 end exponent space plus space 5 straight x to the power of negative 6 end exponent right parenthesis space dx space equals space minus dt
therefore comma
straight I space equals space minus integral dt over straight t cubed space equals space minus integral straight t to the power of negative 3 end exponent space dt
equals negative space fraction numerator straight t to the power of negative 3 space plus space 1 end exponent over denominator negative 3 plus 1 end fraction plus space straight C space equals fraction numerator space 1 over denominator 2 straight t end fraction space plus space straight C
equals space fraction numerator straight x to the power of 10 over denominator 2 left parenthesis straight x to the power of 5 space plus straight x cubed space plus 1 squared right parenthesis end fraction space plus straight C
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2.

Two sides of a rhombus are along the lines, x−y+1=0 and 7x−y−5=0. If its diagonals intersect at (−1, −2), then which one of the following is a vertex of this rhombus?

  • (−3, −9)

  • (−3, −8)

  • (1/3, -8/3)

  • (1/3, -8/3)

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3.

The centres of those circles which touch the circle, x2+y2−8x−8y−4=0, externally and also touch the x-axis, lie on:

  • a circle

  • an ellipse which is not a circle

  • a hyperbola.

  • a hyperbola.

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4.

The integral integral fraction numerator dx over denominator straight x squared space left parenthesis straight x to the power of 4 plus 1 right parenthesis to the power of begin display style 3 over 4 end style end exponent end fraction space equals

  • open parentheses fraction numerator straight x to the power of 4 plus 1 over denominator straight x to the power of 4 end fraction close parentheses to the power of 1 fourth end exponent space plus straight C
  • left parenthesis straight x to the power of 4 plus 1 right parenthesis to the power of 1 fourth end exponent space plus straight C
  • negative left parenthesis straight x to the power of 4 plus 1 right parenthesis to the power of 1 fourth end exponent space plus straight C
  • negative left parenthesis straight x to the power of 4 plus 1 right parenthesis to the power of 1 fourth end exponent space plus straight C
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5.

The integral integral subscript 2 superscript 4 fraction numerator log space straight x squared over denominator log space straight x squared space plus space log space left parenthesis 36 minus 12 space straight x plus straight x squared right parenthesis end fraction dx  is equal to 

  • 2

  • 4

  • 1

  • 1

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6.

The number of points having both coordinates as integers that lie in the interior of the triangle with vertices (0,0), (0,41) and (41,0) is

  • 901

  • 861

  • 820

  • 820

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7.

The integral integral open parentheses 1 plus straight x space minus 1 over straight x close parentheses straight e to the power of straight x plus 1 over straight x end exponent dx is equal to 

  • left parenthesis straight x minus 1 right parenthesis straight e to the power of straight x plus 1 over straight x space plus straight C end exponent
  • xe to the power of straight x plus 1 over straight x end exponent plus straight C
  • left parenthesis straight x plus 1 right parenthesis straight e to the power of straight x plus 1 over straight x space plus straight C end exponent
  • left parenthesis straight x plus 1 right parenthesis straight e to the power of straight x plus 1 over straight x space plus straight C end exponent
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8.

The Integral integral subscript 0 superscript straight pi square root of 1 plus 4 space sin squared straight x over 2 minus 4 sin straight x over 2 dx end root is equal to

  • π-4

  • fraction numerator 2 space straight pi over denominator 3 end fraction minus 4 minus 4 square root of 3
  • 4 square root of 3 minus 4
  • 4 square root of 3 minus 4
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9.

At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production P with respect to the additional number of workers x is given by dP over dx space equals space 100 minus 12 square root of straight x. If the firm employs 25 more workers, then
the new level of production of items is

  • 2500

  • 3000

  • 3500

  • 3500

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10.

If ∫f x dx =Ψ (x),then ∫x5 f(x3)dx is equal to

  • 1 third space open square brackets straight x cubed straight psi left parenthesis straight x cubed right parenthesis minus integral straight x squared space straight psi left parenthesis straight x cubed right parenthesis dx close square brackets space plus straight C
  • 1 third space straight x cubed straight psi left parenthesis straight x cubed right parenthesis minus 3 integral straight x squared space straight psi left parenthesis straight x cubed right parenthesis dx space plus straight C
  • 1 third straight x cubed straight psi left parenthesis straight x cubed right parenthesis minus integral straight x squared space straight psi left parenthesis straight x cubed right parenthesis dx space plus space straight C
  • 1 third straight x cubed straight psi left parenthesis straight x cubed right parenthesis minus integral straight x squared space straight psi left parenthesis straight x cubed right parenthesis dx space plus space straight C
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