- #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.
rootX said:d/dx (log |x-3|) != 1/(x-3)
for x element of (-inf, inf) for sure!
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.
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|.
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.
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.
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.