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Cara Duke

Cara Duke

Answered question

2022-05-25

Evaluate:
x x 3 x 2 + 1 d x
None of the regular methods to integrate rational functions (substitution, partial fractions, etc.) seems to work.

Answer & Explanation

komizmtk

komizmtk

Beginner2022-05-26Added 8 answers

The denominator has only one real root, x 0 . Hence,
x 3 x 2 + 1 = ( x x 0 ) ( x 2 + x ( x 0 1 ) + x 0 2 x 0 ) .
Then, x x 3 x 2 + 1 = A x x 0 + B x + C x 2 + ( x 0 1 ) x + x 0 2 x 0 ,
which can be solved for A,B,C:
A = 1 3 x 0 2 ,
B = A ,
C = A ( x 0 1 ) ,
making the integral approachable.

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