My analysis book makes the following statement: Every rational function with real coefficients can

manierato5h

manierato5h

Answered question

2022-06-05

My analysis book makes the following statement:
Every rational function with real coefficients can be integrated in terms of
1, rational functions,
2. logarithm functions,
3. arctangent functions.
Does it mean that after polynomial division, partial fraction expansion, completing the square, substituting where needed, and knowing that
d x ( x a ) n = 1 n 1 1 ( x a ) n 1       ( n 1 ) d x x a = log | x a | d x x 2 + 1 = arctan x
one can integrate any rational function?

Answer & Explanation

Layla Love

Layla Love

Beginner2022-06-06Added 29 answers

Yes, you need to know about polynomial division, but you also need to know about the degree of the numerator and the denominator, beacuse (and this comes from the link) the degree of the numerator and denominator determines what to do first. If the degree of the numerator is greater than that of the denominator, you do polynomial division, otherwise, you need to factor the denominator. Then you do partial fraction expansion (like you said). Yes, you may need to know the conditions you stated because the rational function could be of any of those forms.
hope this helps!

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