What is the reason for different integrals if equations are the same?

  • Thread starter gladius999
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In summary, the two equations 2/(2x+2) and 1/(x+1) are essentially the same, as the top is a derivative of the bottom in both cases. Therefore, their integrals should also be the same. The additional constant added to the indefinite integrals does not affect the value of the integral, and can be canceled out when finding the definite integral. This concept holds true for any anti-derivatives of the same function.
  • #1
gladius999
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I am a little confused here. If the integral of f'(x)/f(x)= ln|f(x)| +k then say the below equations which are the same give different results?

2/(2x+2)

The top is a derivative of the bottom, so the integral is ln|2x+2|+k

1(x+1)

This is the same as the first equation. The top is also a derivative of the bottom, so the integral is ln|x+1|+k

The two equations are the same so how could they give different integrals?

Thanks for your time
 
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  • #2
ln|2x+2|+k=ln|2(x+1)|+k=ln|x+1|+ln|2+k=ln|x+1|+K, K=k+ln|2|

Thus, the two examples you gave differ only by a constant, something you know that anti-derivatives are allowed to differ with.

Agreed?
 
  • #3
yes i agree it only differs by a constant, but that constant is not counted when finding the definite integral. That means that if u find the definite integral of the equations u would get different answers?
 
  • #4
No you wouldn't, the constant gets canceled when you do definite integration. Try it.
 
  • #5
Remember that ln|a(x+b)|-ln|a(X+b)|=ln(|x+b|/|X+b|), irrespective of the value of "a".

Agreed?
 
  • #6
boboYO said:
No you wouldn't, the constant gets canceled when you do definite integration. Try it.

thats exactly what I am talking about. The constant gets canceled therefore the two integrals must be different which doesn't make sense because they are the essentially the same equation.
 
  • #7
arildno said:
Remember that ln|a(x+b)|-ln|a(X+b)|=ln(|x+b|/|X+b|), irrespective of the value of "a".

Agreed?

Ooo. Yes that makes sense now. So with the different constants added to the indefinite integrals, the two integrals are equal in value?
 
Last edited:
  • #8
Given two anti-derivatives of the same function, you can always make them identical by adding some constant to one of them.

I hope that answers your question.
 
  • #9
yes it does. Thank you very much good sir.
 

FAQ: What is the reason for different integrals if equations are the same?

What is the purpose of integrating f'(x)/f(x)?

The purpose of integrating f'(x)/f(x) is to find the antiderivative or integral of a function that is in the form of a fraction, where the numerator is the derivative of the denominator. This process is often used in calculus and is important in solving various mathematical and scientific problems.

How do you integrate f'(x)/f(x)?

To integrate f'(x)/f(x), you can use the substitution method or the partial fractions method. In the substitution method, you substitute u = f(x) and du = f'(x)dx, which will transform the integral into a simpler form. In the partial fractions method, you express the fraction as a sum of simpler fractions and then integrate each term separately.

What are the common applications of integrating f'(x)/f(x)?

Integrating f'(x)/f(x) has many practical applications in physics, engineering, and economics. For example, it can be used to calculate the work done by a variable force, find the area under a curve, or determine the growth rate of a population. It is also used in solving differential equations, which are commonly used to model a wide range of natural phenomena.

Can you integrate f'(x)/f(x) when f(x) is a complex function?

Yes, f'(x)/f(x) can be integrated even when f(x) is a complex function. However, it may require more advanced techniques such as contour integration or the Cauchy-Riemann equations. It is important to note that these methods may not always yield a closed-form solution and numerical methods may be used instead.

What are the important properties of integrating f'(x)/f(x)?

Some important properties of integrating f'(x)/f(x) include linearity, where the integral of a sum is equal to the sum of the integrals, and the constant multiple rule, where a constant can be factored out of the integral. Additionally, the integral of a function from a to b is equal to the negative of the integral of the same function from b to a, which is known as the reverse rule.

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