I have tried to solve a problem here and I've got an answer but I just don't know if it is the right

Brenden Tran

Brenden Tran

Answered question

2022-06-22

I have tried to solve a problem here and I've got an answer but I just don't know if it is the right one.
The problem is: For what conditions
a x 2 + b x + c x 3 ( x 1 ) 2 d x
represents a rational function.
First I have calculated the integral and the result is:
( a + 2 b + 3 c ) ln | x | 2 c + b x ( a + 2 b + 3 c ) ln | x 1 | a + b + c x + 1 c 2 x 3 + K
where K is a constant.
From here I have taken the conditions
( a + 2 b + 3 c = 0 )  and  ( ( b + 2 c ! = o )  or  ( a + b + c ! = 0 ) )
which is the same as
( a = 2 b 3 c )  and  ( b ! = 2 c )
So, from my result, the condition above must be satisfied in order for that function to be a rational function.
P.S != means not equal to
Can anyone tell whether the solution is correct or not and if it is not then help me with it ?

Answer & Explanation

Dustin Durham

Dustin Durham

Beginner2022-06-23Added 31 answers

The only condition you want is that the coefficient in front of the logarithms are zero, the rest is considered to be a rational function (also, a polynomial is considered to be a rational function). In your case, this means that
a + 2 b + 3 c = 0.
(The term ( a + b + c ) / ( x + 1 ) should be ( a + b + c ) / ( 1 x ), but that does not change the answer.)
Mohamed Mooney

Mohamed Mooney

Beginner2022-06-24Added 5 answers

Since
1 x 3 ( x 1 ) 2 = 1 ( x 1 ) 2 3 x 1 + 3 x + 2 x 2 + 1 x 3 ,
it follows that the integral of ( a x 2 + b x + c ) over x 3 ( x 1 ) 2 is rational iff, in the integrand function, the coefficient of 1 / x, that is a + 2 b + 3 c, and the coefficent of 1 / ( x 1 ), that is ( a + 2 b + 3 c ) are zero.
So the condition should be simply a + 2 b + 3 c = 0

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