Confused about integral of f'(x)/f(x)

So ln(3x^2)=2ln(x)+ln(3), and ln(3) is just a constant, so the two expressions are essentially the same except for the constant term. This is why they are not equal.
  • #1
Lucy Yeats
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0
A fraction obviously stays the same if you multiply top and bottom by the same constant. If you integrate two equal functions, surely you should get the same answer? By applying the rule in the title, you can get different answers. I'm very confused.

Example: 2x/x^2=6x/3x^2
The integral of 2x/x^2 is ln(x^2).
The integral of 6x/3x^2 is ln(3x^2)

How come these aren't equal? Is it to do with the +c?
 
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  • #2
Remember that [itex]\ln (ab) = \ln a + \ln b[/itex]. Hence [itex]\ln(3x^2)=\ln(x^2)+\ln(3)[/itex], so the two expressions just differ by a constant.
 
  • #3
Ah, okay. Thanks!
 
  • #4
Lucy Yeats said:
A fraction obviously stays the same if you multiply top and bottom by the same constant. If you integrate two equal functions, surely you should get the same answer? By applying the rule in the title, you can get different answers. I'm very confused.

Example: 2x/x^2=6x/3x^2
The integral of 2x/x^2 is ln(x^2).
The integral of 6x/3x^2 is ln(3x^2)

How come these aren't equal? Is it to do with the +c?

Yes, it is about the plus C.
ln(3x^2)=ln(3)+ln(x^2)
 
  • #5


It is important to understand that when we integrate a function, we are essentially finding the antiderivative of that function. This means that we are finding a function whose derivative is equal to the original function. Therefore, when we integrate different functions, we may end up with different answers because they have different derivatives.

In the example given, although the two functions may look similar, they are not the same. The first function, 2x/x^2, simplifies to 2/x, while the second function, 6x/3x^2, simplifies to 2/x. These two functions may have the same value at certain points, but they are not equivalent.

When we integrate 2/x, we get ln(x) + c, where c is a constant. Similarly, when we integrate 2/x, we get ln(3x) + c. The constants may be different, but that does not change the fact that these are two different functions.

Furthermore, it is important to note that when we multiply the top and bottom of a fraction by the same constant, we are essentially changing the function and its derivative. In the case of integrating, this means we are finding the antiderivative of a different function, which may result in a different answer.

So, to answer your question, the reason why these integrals are not equal is because the functions being integrated are not the same, and the constants that are added during the integration process may also be different. It is not related to the +c, but rather the fact that the functions being integrated are not identical. I hope this helps to clarify any confusion you may have.
 

1. What is the definition of an integral?

The integral of a function f(x) is a mathematical operation that calculates the area under the curve of the function over a specific interval. It is represented by the symbol ∫ and is the inverse operation of differentiation.

2. How do you solve an integral?

To solve an integral, you need to first identify the function and the limits of integration. Then, you can use various integration techniques such as substitution, integration by parts, or partial fractions to find the anti-derivative of the function. Finally, you can evaluate the integral using the limits of integration to find the numerical value.

3. What is the difference between definite and indefinite integrals?

A definite integral has specific limits of integration and gives a numerical value as the result. An indefinite integral does not have limits of integration and gives a function as the result, also known as the anti-derivative.

4. How does the fundamental theorem of calculus relate to integrals?

The fundamental theorem of calculus states that the derivative of an integral is equal to the original function. In other words, integration is the inverse of differentiation. This theorem is crucial in solving integrals, as it allows for the use of anti-derivatives to find the numerical value of an integral.

5. What is the significance of the natural logarithm in the integral of f'(x)/f(x)?

The natural logarithm, ln(x), is often used in the integral of f'(x)/f(x) because it is the anti-derivative of 1/x. This means that when solving an integral that contains a function in the form of 1/x, the natural logarithm can be used to find the anti-derivative. Additionally, ln(x) is often used in the integration by parts technique to simplify the integral.

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