Evaluate Integral using USubstitution

Evaluate $\displaystyle \int_{0}^{3} e^{2x} \, dx$ using the $u$-substitution method, where $u = 2x$.

The integral $\int_{0}^{3} e^{2x} \, dx$ presents a scenario ideal for applying the technique known as u-substitution, a method commonly used to simplify integration problems. When faced with an integral involving a composite function, substituting part of the function to reduce it to a more familiar form can be an effective strategy. In this problem, the substitution $u = 2x$ simplifies the integral by transforming the exponent of $e$ into a linear form, which is easier to integrate.

U-substitution is analogous to the chain rule used in differentiation but in reverse. It systematically reduces the complexity of the problem by identifying a part of the function to substitute, thereby transforming an intimidating function into a manageable one. The process involves first identifying the substitution variable (here, $u = 2x$), finding the corresponding differential ($du = 2\,dx$), and rewriting the integral in terms of $u$. This often results in an integral that is straightforward to evaluate.

After integrating with respect to the new variable, don't forget the final step: substituting back to the original variable in the context of the given limits. This ensures the solution is expressed appropriately for the original problem. Mastering this technique not only aids in solving integrals directly but also enhances problem-solving skills for more complex calculus problems, fostering a deeper understanding of the underlying mechanics of integration.

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