Why is Anti Derivative of (1/x-3) Equal to -ln|x+3|?

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In summary, the anti derivative of 1/(x-3) is -ln|x+3| and not ln|x-3|. This is because the latter only defines the differentiation function for positive x values, while the former works for all x values except for x=3. This is due to the absolute value in the function, which accounts for negative x values as well. However, there was a typo in the conversation, as the derivative of ln|x-3| is not equal to 1/(x-3) for all x values.
  • #1
elitespart
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Why is the anti derivative of 1/(x-3) equal to -ln[tex]\left|x+3\right|[/tex]. Why isn't it just ln[tex]\left|x-3\right|[/tex]. Does it have something to do with the absolute value? Thanks.
 
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  • #2
for negative x?

using just log |x-3| defines the differentiation function for pos x only
 
  • #3
it's not -ln|x+3| and is in fact ln|x-3| . Take the derivative of your first answer and second and see which one works.
 
  • #4
Alright so I'm guessing there was a typo. Thanks for your help.
 
  • #5
d/dx (log |x-3|) != 1/(x-3)
for x element of (-inf, inf) for sure! edit: wrong

ooops.. my mistake. I was ignoring the absolute part
 
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  • #6
rootX said:
d/dx (log |x-3|) != 1/(x-3)
for x element of (-inf, inf) for sure!

For [itex]x \neq 3[/itex]
 
  • #7
the original function is undefined for x=3 also btw.
 

1. What is an anti-derivative?

An anti-derivative is the reverse process of differentiation, where we find the original function that was differentiated. It is also known as the indefinite integral.

2. Why is the anti-derivative of (1/x-3) equal to -ln|x+3|?

The anti-derivative of a function is not unique, as it can have different constants of integration. However, for the function (1/x-3), the constant of integration is -3, which results in the solution -ln|x+3|.

3. Can you explain the notation used in the anti-derivative -ln|x+3|?

The notation ln|x| represents the natural logarithm of the absolute value of x. The absolute value is used because the natural logarithm function is only defined for positive numbers.

4. How do you verify that the anti-derivative of (1/x-3) is -ln|x+3|?

To verify this, we can take the derivative of -ln|x+3|, which is 1/(x+3). This is the original function, confirming that -ln|x+3| is indeed the anti-derivative.

5. Can you give an example of how the anti-derivative of (1/x-3) is used in real-life applications?

The anti-derivative of (1/x-3) is used in the field of physics, specifically in the study of electric fields and potential energy. It is also used in economics to calculate the marginal cost function.

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