Integrating f'(x)/f(x)

[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 cancelled 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 cancelled when you do definite integration. Try it.

thats exactly what im talking about. The constant gets cancelled therefore the two integrals must be different which doesnt 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.