Use any method to evaluate the integrals in Exercises 55–66.
∫ 2^x / (2²x + 2^x - 2) dx

Verified step by step guidance

1. First, rewrite the integral to clearly identify the expressions involved: \( \int \frac{2^x}{2^{2x} + 2^x - 2} \, dx \). Notice that \(2^{2x} = (2^x)^2\), so let’s set \( t = 2^x \) to simplify the integral.
2. Express \( dx \) in terms of \( dt \). Since \( t = 2^x \), take the natural logarithm derivative: \( \frac{dt}{dx} = 2^x \ln(2) = t \ln(2) \), which implies \( dx = \frac{dt}{t \ln(2)} \).
3. Substitute \( t \) and \( dx \) into the integral: \( \int \frac{t}{t^2 + t - 2} \cdot \frac{dt}{t \ln(2)} = \int \frac{1}{(t^2 + t - 2) \ln(2)} \, dt \). The \( t \) in numerator and denominator cancels out.
4. Focus on the integral \( \int \frac{1}{t^2 + t - 2} \, dt \). Factor the quadratic denominator: \( t^2 + t - 2 = (t + 2)(t - 1) \). Use partial fraction decomposition to express \( \frac{1}{(t + 2)(t - 1)} \) as \( \frac{A}{t + 2} + \frac{B}{t - 1} \).
5. Solve for constants \( A \) and \( B \), then integrate each term separately: \( \int \frac{A}{t + 2} \, dt + \int \frac{B}{t - 1} \, dt \). After integrating, substitute back \( t = 2^x \) to express the answer in terms of \( x \).

Key Concepts

Integration Techniques

Integration techniques include various methods such as substitution, integration by parts, partial fractions, and recognizing standard integral forms. Choosing the right technique simplifies the integral and makes it solvable. For complex expressions involving exponents, substitution is often useful to transform the integral into a more manageable form.

Properties of Exponential Functions

Exponential functions like 2^x have unique properties, such as the rule 2^{a+b} = 2^a * 2^b, which help simplify expressions. Understanding how to manipulate and rewrite exponential terms is crucial for simplifying the integrand and identifying substitution candidates.

Algebraic Simplification and Substitution

Algebraic simplification involves rewriting the integrand to a simpler form, often by factoring or combining like terms. Substitution replaces a complicated expression with a single variable, making the integral easier to evaluate. Recognizing patterns in the denominator and numerator can guide effective substitution choices.