Antiderivative of a polynomial fraction I am trying to solve this problem and I can't seem to under

Brock Byrd

Brock Byrd

Answered question

2022-07-02

Antiderivative of a polynomial fraction
I am trying to solve this problem and I can't seem to understand where I am going wrong. My understanding of antiderivatives is this formula:
x n d x = x n + 1 n + 1 + c
The problem I am trying to solve is:
f ( x ) = x 5 x 3 + 8 x x 4
I think there are two possible parts that I am getting wrong, I started the problem by separating them into individual fractions and solving each fraction at a time. I think I get it wrong at either the ln(x) or 8 x 2 2 . Here is my attempt: f ( x ) = x 5 x 4 x 3 x 4 + 8 x x 4 = x 1 x + 8 x 3
Then from here I get the antiderivative:
F ( x ) = x 2 2 ln ( x ) + 8 x 2 2 + c = x 2 2 ln ( x ) 4 x 2 + c
I keep getting the wrong answer when I plug it in, am I missing some fundamental piece of information about antiderivatives which is causing me to mess up the ln(x) or is my algebra wrong and I mess up around 8 x 2 2 ?

Answer & Explanation

postojahob

postojahob

Beginner2022-07-03Added 13 answers

Explanation:
The domain of the natural logarithm is ( 0 , ). However, 1/x is defined on ( , 0 ) ( 0 , ), so should be the derivative of some function defined on this larger set. That "some function" is ln|x|. That is d d x ln | x | = 1 x , x 0 so ln | x | + C = 1 x d x , x 0 .
cooloicons62

cooloicons62

Beginner2022-07-04Added 4 answers

Explanation:
The problem is in the integration of x:
x 1 x + 8 x 3 d x = x 2 2 ln x 4 x 2 + C

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