Evaluating Improper Integrals?

📌 CONCEPT: An improper integral is a limit of a definite integral as the upper limit of integration approaches infinity, used to evaluate the area under a curve when the function is unbounded or has infinite discontinuities.

📐 RULE / FORMULA: The improper integral ∫[0 to ∞] e^(-x) / (x^2 + 1) dx can be evaluated by first finding the antiderivative of the function f(x) = e^(-x) / (x^2 + 1) and then applying the limit as the upper limit of integration approaches infinity.

💡 WORKED EXAMPLE: Let's evaluate ∫[0 to ∞] e^(-x) / (x^2 + 1) dx. We start by finding the antiderivative of f(x) = e^(-x) / (x^2 + 1) using the method of substitution. After finding the antiderivative, we apply the limit as the upper limit of integration approaches infinity.

⚠️ COMMON MISTAKE: Students often forget to check the convergence of the integral by applying the limit, or they may incorrectly apply the antiderivative formula, leading to incorrect results.