Integrate ∫ (e^2x - 1) / (e^2x + 1) dx.

To solve the integral ∫ (e^(2x) - 1) / (e^(2x) + 1) dx, we can use a substitution method. Let's start by simplifying the expression.

Substitution Step

Let \( u = e^{2x} \). Then, the differential \( du = 2e^{2x} dx \) or \( dx = \frac{du}{2u} \).

Rewriting the Integral

Now, we can rewrite the integral in terms of \( u \):

• Substituting \( e^{2x} \) with \( u \), we have:
• Numerator: \( u - 1 \)
• Denominator: \( u + 1 \)

The integral becomes:

∫ (u - 1) / (u + 1) * (1/(2u)) du

Breaking Down the Integral

This can be split into two parts:

∫ (1/2) * (1 - 2/(u + 1)) du

Integrating Each Part

Now, we can integrate each term separately:

• ∫ (1/2) du = (1/2)u
• ∫ (-1) du = -ln|u + 1|

Combining Results

Putting it all together, we have:

(1/2)u - ln|u + 1| + C

Back Substitution

Now, substitute back \( u = e^{2x} \):

(1/2)e^{2x} - ln|e^{2x} + 1| + C

Final Answer

The final result of the integral is:

(1/2)e^{2x} - ln(e^{2x} + 1) + C