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

[gladius999]

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

 

[arildno]

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?

 

[gladius999]

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?

 

[boboYO]

No you wouldn't, the constant gets canceled when you do definite integration. Try it.

 

[arildno]

Remember that ln|a(x+b)|-ln|a(X+b)|=ln(|x+b|/|X+b|), irrespective of the value of "a".

Agreed?

 

[gladius999]

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.

 

[gladius999]

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?

 

[arildno]

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.

 

[gladius999]

yes it does. Thank you very much good sir.