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Integration problem (algebraic+trigonometric function)

  • #1
Krushnaraj Pandya
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Homework Statement


find integral of the function- (x^2 + cos^2)(cosec^2) / (1+x^2)

2. The attempt at a solution
I noticed the denominator is the derivative of Arctan(x), I tried integrating by parts with various choices for 1st and second function but all of them end up being more complicated, I tried factoring in sec^2(x) which also didn't go a long way. I would appreciate some help to proceed further with this integral
 

Answers and Replies

  • #2
vela
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  • #3
Krushnaraj Pandya
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What does this mean?
It means I tried multiplying and dividing by sec^2(x), but still couldn't integrate it
 
  • #4
vela
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You're overthinking it. Multiply the numerator out and go from there.
 
  • #5
Krushnaraj Pandya
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You're overthinking it. Multiply the numerator out and go from there.
you mean separating numerator and denominator as two different functions and then integrate by parts? I already tried that
 
  • #6
vela
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No, you just want to simplify the integrand first.
 
  • #7
Krushnaraj Pandya
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No, you just want to simplify the integrand first.
what did you mean by "multiply the numerator out?" multiply cosec^2 inside?
 
  • #8
vela
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Distribute the factor of ##csc^2 x## into the sum.
 
  • #9
Krushnaraj Pandya
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Distribute the factor of ##csc^2 x##.
done already. ## cosec^2(x) x^2 + cot^2(x) ## in the numerator
then i separated both terms but i'd have to integrate both of them by parts which is really long
 
  • #10
vela
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You haven't really simplified the integrand. To me, the obvious thing to try here is a trig identity. You have csc^2 and cot^2...a natural choice comes to mind. See what happens.
 
  • #11
Krushnaraj Pandya
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You haven't really simplified the integrand. To me, the obvious thing to try here is a trig identity. You have csc^2 and cot^2...a natural choice comes to mind. See what happens.
It was so easy! I really did overthink it. Thanks a lot though :)
 

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