Two different results for the same integral

[fredgarvin22]

hello everybody

i hope i make this clear and to the point. there is an integral that is bothering me. i will express it as the combination of 2 integration formulas that you can look up, under logarithmic functions (http://en.wikipedia.org/wiki/List_of_integrals_of_logarithmic_functions) . here are two identities from that list:

$$\int \ln(cx) dx = x\ln(cx) - x$$
and
$$\int \ln(ax + b) dx = x\ln(ax + b) - x + (b/a)\ln(ax + b)$$

I have an equation(it actually comes from a famous paper in physics) that basically represents the difference between the two. To make things simpler, a = c = 1 in my equation.
So I have:

(1) $$\int \ln(x + b) dx - \int \ln(x) dx$$

using the identities I have:

$$x\ln(x + b) - x + (b)\ln(x + b) - x\ln(x) + x$$
$$x\ln(x + b) + (b)\ln(x + b) - x\ln(x)$$

$$(x + b)\ln(x + b) - x\ln(x)$$

so that's that. Now let me do it slightly differently(and don't ask why):

I will factor out the 'b' from the equation first (1):

(2) $$\int \ln[(x/b + 1)*(b)] dx - \int \ln[(x/b)*b] dx$$

$$\int \ln[(x/b + 1) + ln(b)] dx - \int \ln[(x/b) + ln(b)] dx$$

$$\int \ln(x/b + 1) dx - \int \ln(x/b)dx + \int ln(b)dx - \int ln(b)dx] dx$$

$$\int \ln(x/b + 1) dx - \int \ln(x/b) dx$$

using the same identities

$$x\ln(x/b + 1) - x + b\ln(x/b + 1) - x\ln(x/b) + x$$
$$x\ln(x/b + 1) + b\ln(x/b + 1) - x\ln(x/b)$$
$$(x + b)\ln(x/b + 1) - x\ln(x/b)$$

this is now a different result from approach (1). You can't, unless I'm wrong, recover it again by resubstituting the factor b in again.

My question: why does this simple factoring out, change the answer here?

 

[mathman]

Constant of integration: (x+b)ln(b)-xln(b) is constant [= (b)ln(b)]. Add it to your second result, you will get the first result.

 

[fredgarvin22]

that's very clever mathman - how did you come up with that?

 

[Gib Z]

He just noticed what the difference between the 2 solutions were. If it ever happens, and its a constant, you know what you forgot.

 

[fredgarvin22]

you are correct, the difference between the solutions is indeed b*ln(b). to me that means that in order for the constant of integration(which could any constant) to be equivilent in both equations, I need to either add or subtract this constant, which ever is appropriate, to the result.

I'm not sure what I have learned here, except that the answers are off only by a constant(b*(ln(b)). perhaps if i think about it for a while it will sink in. i'd still like to see mathman elaborate just a little more on his answer, because my math is so rusty and terrible.

in any case thank both of you guys for taking the time to answer.

 

[arildno]

Why should they be equivalent?
Why should one technique that gives you one anti-derivative give you the same anti-derivative as using some other?

DEFINITE integrals are unique, anti-derivatives are not.

 

[fredgarvin22]

i expected them to be equivelant because the (1) and (2) look equivelant to me. if you did this trick on just one of the identities alone, then the answer comes out the same. it seems to come up almost by accident because 2 terms from the two identities cancel each other out. and evidently this is buried somewhere in the integration constant.

 

[mathman]

I'm not sure what I have learned here, except that the answers are off only by a constant(b*(ln(b)). perhaps if i think about it for a while it will sink in. i'd still like to see mathman elaborate just a little more on his answer, because my math is so rusty and terrible.

My starting point was the observation that the integral of ln(b) is (x)ln(b).
Since we are dealing with indefinite integrals, the constant is arbitrary, (b)ln(b) is perfectly acceptable.