Final answer:
To solve the given integral, we can use partial fractions to break down the integrand into simpler fractions and then integrate those separately.
Step-by-step explanation:
To find the indefinite integral ∫ e^x / ((e^(2x) + 1)(e^x - 1)) dx, we can start by using partial fractions to break down the integrand into simpler fractions. The denominator can be factored as (e^x - 1)(e^x + 1), so we can write the integrand as A/(e^x - 1) + B/(e^x + 1). Now we need to find the values of A and B.
Using the method of substitution, let's substitute e^x = u for the first fraction, and e^x = -u for the second fraction. We get A/u + B/-u. Multiplying both sides by (e^x - 1)(e^x + 1), we have e^x * (A/u) + e^x * (B/-u) = A * (e^x + 1) - B * (e^x - 1). Now we can substitute e^x = u and solve for A and B.
After finding the values of A and B, we can now integrate the two separate fractions. The integral of A/(e^x - 1) is A * ln|e^x - 1| + C, where C is the constant of integration. The integral of B/(e^x + 1) is B * ln|e^x + 1| + D, where D is another constant of integration. Adding these two integrals together gives us the final result.