7–84. Evaluate the following integrals.
33. ∫ [eˣ / (a² + e²ˣ)] dx, where a ≠ 0

Verified step by step guidance

1. Step 1: Recognize that the integral involves a rational function with an exponential term in the numerator and a sum of squares in the denominator. This suggests a substitution method might simplify the problem.
2. Step 2: Let u = eˣ. Then, differentiate u with respect to x to find du: du = eˣ dx. This substitution replaces eˣ dx in the integral with du.
3. Step 3: Rewrite the integral in terms of u. Since u = eˣ, the denominator becomes a² + u², and the integral becomes ∫ [u / (a² + u²)] du.
4. Step 4: Recognize that the new integral ∫ [u / (a² + u²)] du can be simplified using the derivative of the denominator. Specifically, the derivative of a² + u² with respect to u is 2u, which suggests a natural logarithm substitution.
5. Step 5: Apply the substitution and simplify the integral. The result will involve a logarithmic function, specifically ln(a² + u²), and then substitute back u = eˣ to express the solution in terms of x.

Key Concepts

Integration Techniques

Integration techniques are methods used to evaluate integrals, which can include substitution, integration by parts, and partial fractions. In this case, recognizing the form of the integrand, particularly the presence of the exponential function and the polynomial in the denominator, is crucial for selecting the appropriate technique to simplify the integral.

Exponential Functions

Exponential functions, such as eˣ, are functions of the form f(x) = a^x, where 'a' is a constant. They have unique properties, including the fact that their derivatives and integrals are also exponential functions. Understanding how to manipulate these functions is essential for evaluating integrals involving them, especially when combined with other algebraic expressions.

Improper Integrals

Improper integrals occur when the limits of integration are infinite or when the integrand approaches infinity within the interval of integration. In this problem, recognizing whether the integral converges or diverges is important, as it affects the evaluation process. Techniques such as limits can be applied to handle these cases effectively.