Problem 40 - Essential Calculus

Question

State (if possible) the method or integration formula you would use to find the antiderivative. Explain why you chose that method or formula. Do not integrate. $$ \int e^{2 x} \sqrt{e^{2 x}+1} d x $$

Step-by-Step Solution

Answer

The appropriate method appears to be the hyperbolic trigonometric substitution, where one would use \( x = sinh(t)\). This could potentially simplify the integrand and allow the integral to be solved. The reasoning behind is the presence of the \( e^{2x} + 1 \) under a square root sign, which is similar to the hyperbolic identity \( cosh^2(t) - sinh^2(t) = 1 \).

Step 1: Identify the integrand

The first step is to identify the integrand. Here the integrand is \( e^{2x} \sqrt{e^{2x}+1} \).

Step 2: Determine suitable method

When looking at the integrand, it becomes clear that neither basic integration formulae nor common integration techniques like substitution can be directly applied due to the complexity of the integrand. However, trigonometric substitution may work. For example, hyperbolic substitution might be a good try. Why? Because hyperbolic functions are helpful when the given integral contains terms of the form \(a^2 + x^2\). In the case of this exercise, taking the hyperbolic sine function, i.e., \( x = sinh(t) \), can be useful.