Losing my mind over a very simple integration problem

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anon_needs_help

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I'm trying to solve the following indefinite integral:

[MATH] \int \frac{1}{5x-1} \,dx [/MATH]
This has a fairly simple solution via u-substitution:

[MATH]u=5x-1[/MATH][MATH]du=5dx[/MATH][MATH] \int \frac{1}{5x-1} \,dx = \frac{1}{5} \int \frac{1}{u} \,du = \frac{1}{5} ln(|u|) + C = \frac{1}{5} ln(|5x-1|) + C [/MATH]
Seems pretty simple, right? However, there's a problem. If I solve it by factoring out [MATH]\frac{1}{5}[/MATH] first, I get a different answer:

[MATH] \int \frac{1}{5x-1} \,dx = \frac{1}{5} \int \frac{1}{x-\frac{1}{5}} \,dx [/MATH]
[MATH]u=x - \frac{1}{5}[/MATH][MATH]du=dx[/MATH]
[MATH]\frac{1}{5} \int \frac{1}{x-\frac{1}{5}} \,dx = \frac{1}{5} \int \frac{1}{u} \,du = \frac{1}{5} ln(|u|) + C = \frac{1}{5} ln(|x-\frac{1}{5}|) + C [/MATH]
Why do these two seemingly both valid methods of solving this basic integral yield different answers? As far as I'm aware, the first one is the correct result. However, I'm not so much interested in the actual answer. I simply want to understand what's wrong with the second method, and why it's giving me a different result.
 
anon_needs_help said:
I'm trying to solve the following indefinite integral:

[MATH] \int \frac{1}{5x-1} \,dx [/MATH]
This has a fairly simple solution via u-substitution:

[MATH]u=5x-1[/MATH][MATH]du=5dx[/MATH][MATH] \int \frac{1}{5x-1} \,dx = \frac{1}{5} \int \frac{1}{u} \,du = \frac{1}{5} ln(|u|) + C = \frac{1}{5} ln(|5x-1|) + C [/MATH]
Seems pretty simple, right? However, there's a problem. If I solve it by factoring out [MATH]\frac{1}{5}[/MATH] first, I get a different answer:

[MATH] \int \frac{1}{5x-1} \,dx = \frac{1}{5} \int \frac{1}{x-\frac{1}{5}} \,dx [/MATH]
[MATH]u=x - \frac{1}{5}[/MATH][MATH]du=dx[/MATH]
[MATH]\frac{1}{5} \int \frac{1}{x-\frac{1}{5}} \,dx = \frac{1}{5} \int \frac{1}{u} \,du = \frac{1}{5} ln(|u|) + C = \frac{1}{5} ln(|x-\frac{1}{5}|) + C [/MATH]
Why do these two seemingly both valid methods of solving this basic integral yield different answers? As far as I'm aware, the first one is the correct result. However, I'm not so much interested in the actual answer. I simply want to understand what's wrong with the second method, and why it's giving me a different result.
Click to expand...
What do you get:

ln [|x-(1/5)|] - ln|(|5x-1|) = ?
 
[MATH]\dfrac{1}{5} \ln|5x-1| + C_1 = \dfrac{1}{5} \ln\left|5\left(x-\frac{1}{5}\right)\right| +C_1 = \dfrac{1}{5} \bigg[\ln{5} + \ln\left|x-\frac{1}{5}\right| \bigg] + C_1 = {\color{red}\frac{1}{5} \ln{5}} + \frac{1}{5} \ln\left|x - \frac{1}{5}\right| + {\color{red}C_1} = \frac{1}{5}\ln\left|x-\frac{1}{5}\right| + {\color{red}C_2}[/MATH]
 
Nothing wrong with your both methods. In the integration world if you get more than one answer and they only differ by a constant, then both of them is a valid anti- derivative.

[MATH]\frac{1}{5}\ln|x - \frac{1}{5}| + C = \frac{1}{5}\ln|\frac{5x}{5} - \frac{1}{5}| + C = \frac{1}{5}\ln|\frac{5x - 1}{5}| + C = \frac{1}{5}\ln|5x - 1| -\frac{1}{5}ln|5| + C = \frac{1}{5}\ln|5x - 1| + C[/MATH]
 
anon_needs_help said:
Thank you!!!

I saw that they were different values, but I didn't realize that the difference is a constant value : [MATH]ln(5)[/MATH]
Click to expand...
Whenever I get into conundrum like this - get two different answers (visually) for the same problem - I look for the real difference in those answers.
 
That integral is complicated compared to this one.
[math] \int(x-3)^2dx[/math]You can square out x-3 or make a simple substitution u = x-3. The answers will look different but differ by a constant.
 
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