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Integrals

Question
CBSEENMA12036014
Wired Faculty App

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 negative straight x to the power of 10 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

Solution

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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