anon_needs_help
New member
- Joined
- Mar 28, 2021
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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.
[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.