Question

the integral of ∫{ 2x^12 + 5x^9 }dx / (x^5 +x³ +1)^2

Answer 1

Hello pari :-) 

Let I = > ∫{ 2x^12 + 5x^9 }dx / (x^5 +x³ +1)^3

I = > ∫ { 2x^12 + 5x^9 }dx  / x^15 ( 1+ x^-2  +x^-5  )³ 

I = > ∫  {2x^-3  + 5 x^-6 }dx /( 1+x^-2 + x^-5 )³

now put 1+x^-2 + x^-5 = > t

(-2x^-3 - 5x^-6 )dx = >  dt 

I = >  ∫dt / t³  =>  - 1/2t² = > -1/2( 1+x^-2 + x^-5 )²   + c is our required answer 

Hopes this will help you !

@ engineer gopal khandelwal