Integrating $e^x / (e^x - 2)$

As others have said, the result was correct but you took the long way to get there. Where could you have saved time.

$u = e^x - 2$ is a good substitution

$du = e^x dx$

here you could have saved a few steps:

$\int \frac {e^x}{e^x - 2} \ dx = \int \frac {(e^x dx)}{e^x - 2} = \int \frac {du}{u}$

And you could even drop the intermediate step.

It is cleaner than what you have.


Your solution is correct. Notice that your integral is of the form: $$\int \frac{f'(x)}{f(x)}~dx=\ln|f(x)|+C$$ So you can easily obtain the solution if you know this fact.

The online portal may have wanted you to omit the absolute value signs as @Ian suggested or it may be due to the brackets you've added on $\ln(|f(x)|)+C$.

Tags:

Integration

Calculus

Exponential Function

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