Integral of exponential and square root using substitution

Evaluate the integral $\displaystyle \int \left( e^x \, \cdot \, \sqrt{e^{2x} - 4}\right) \, dx$ using substitution.

This problem involves evaluating an integral that combines exponential functions and square roots, specifically $e^x$ and $\sqrt{e^{2x} - 4}$. The challenge here lies in simplifying the expression under the square root to enable the use of substitution. Substitution is a powerful technique in calculus that simplifies complex integrals by transforming the variables, effectively reducing the integral into a form that is easier to solve. Understanding the relationships between the expressions and their derivatives is crucial in choosing the right substitution.

In this problem, you'll need to consider how to express $e^{2x} - 4$ in a form that allows for straightforward substitution. Generally, problems like this require looking for a substitution that can simplify the square root or allow the exponential function to combine with it nicely. Once the substitution is chosen, it is typically necessary to also substitute $dx$, requiring knowledge of derivatives and how they transform under substitution.

This kind of problem hones skills in recognizing patterns that lend themselves to substitution, manipulating exponential expressions, and applying derivative transformations. While it may seem daunting initially due to the unusual combination of exponential and square root functions, with practice, identifying useful substitutions becomes more intuitive. Such problems are excellent for reinforcing the student's ability to navigate through complex expressions and enhance their integral solving strategy.

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