# Integrating f'(x)/f(x)

1. Oct 19, 2009

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?

2. Oct 19, 2009

### 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?

3. Oct 19, 2009

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. Oct 19, 2009

### boboYO

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

5. Oct 19, 2009

### arildno

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

Agreed?

6. Oct 19, 2009

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.

7. Oct 19, 2009

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: Oct 19, 2009
8. Oct 19, 2009

### arildno

Given two anti-derivatives of the same function, you can always make them identical by adding some constant to one of them.

9. Oct 19, 2009