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sir / mam solve this question take refrence from image i have uploaded

sir / mam solve this question take refrence from image i have uploaded

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Grade:12th pass

1 Answers

Aditya Gupta
2081 Points
3 years ago
we wish to integrate (1+x^2)/(4x^4 – 4sqrt2*x + 3)
note that 4x^4 – 4sqrt2*x + 3= (2x^2+2sqrt2x+3)(2x^2 – 2sqrt2x+1)
once we see this factorisation, the ques becomes easy.
also, note that the sum of factors 2x^2+2sqrt2x+3 and 2x^2 – 2sqrt2x+1 is 4(x^2+1)
hence, (1+x^2)/(4x^4 – 4sqrt2*x + 3)= [1/(2x^2+2sqrt2x+3) + 1/(2x^2 – 2sqrt2x+1)]/4
= [1/(4x^2+4sqrt2x+6) + 1/(4x^2 – 4sqrt2x+2)]/2
now write 4x^2 – 4sqrt2x+2= (2x – sqrt2)^2
and 4x^2+4sqrt2x+6= (2x+sqrt2)^2 + (2)^2
after that the integration of these 2 quantities becomes extremely easy since they turn into std forms.
KINDLY APPROVE :))

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