Calculus: Integral of (e^x + 1) / (e^x - 1) - Step-by-Step Solution

Published on May 21, 2025

Calculus

This calculus problem involves evaluating the integral of (e^x + 1) / (e^x - 1) dx. The solution uses substitution and partial fraction decomposition to find the antiderivative.

Question

Answer

The problem asks to evaluate the integral:

We can use a substitution method. Let

. Then

. This means

Substitute these into the integral:

Now, we can use partial fraction decomposition for the integrand

.
Let

Multiply both sides by

To find

, set

To find

, set

So, the partial fraction decomposition is:

Now, integrate term by term:

Finally, substitute back

Since

Alternatively, we can manipulate the integrand directly:

So the integral becomes:

For the second integral, let

. Then

. So

Using partial fractions:

Set

.
Set

.
So,

Substitute back

Now substitute this back into the main integral:

Both methods yield the same result.

The final answer is

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