91Ó°ÊÓ

Whenever we encounter an integral that seems complex, one handy technique that often helps is the **u-substitution**. This approach simplifies the integral by substituting a part of the integrand with a single variable, commonly denoted as \( u \). By effectively reducing the complexity of the integral, we can focus on more straightforward functions.
\(\)In the exercise given, \( e^x \) was a natural choice for \( u \) because it's present both inside the cosine function and alongside \( dx \). When you let \( u = e^x \), the corresponding differential \( du \) becomes \( e^x \, dx \) which transforms the integral into \( \int \cos(u) \, du \).
\(\)This results in a simpler integral where the more complicated exponential function is replaced by \( u \). This substitution step is crucial because it sets the stage for easier integration by dealing with standard integrals, like \( \int \cos(u) \, du \), which are more manageable.

An **antiderivative** is a function whose derivative yields the original function. In the context of integration, finding the antiderivative means reversing the differentiation process to identify a function whose derivative corresponds to the integrand given in the problem.
\(\)After we've simplified the integral using u-substitution, the next task is to determine the antiderivative of the new integrand. For our exercise, the problem simplifies to finding the antiderivative of \( \cos(u) \).
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• The antiderivative of \( \cos(u) \) is \( \sin(u) \).
• This result is derived from the basic derivative rules, where \( \sin(u) \) differentiates back to \( \cos(u) \).

\(\)So, the integral \( \int \cos(u) \, du \) results in \( \sin(u) \), symbolizing the process is complete for the new integral setup. This simplified step highlights the power of understanding and using antiderivative rules to navigate through integration problems.

When finding indefinite integrals, the **constant of integration** always emerges as a key component. It represents an undetermined constant, denoted usually by \( C \), which accounts for the family of all possible antiderivatives.
\(\)Whenever you evaluate an indefinite integral, such as our \( \int \cos(u) \, du \), you should append \( C \) to the result to reflect that differing values of \( C \) cover all functions sharing the same slope, extending indefinitely along the vertical axis in the Cartesian coordinate system.
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• This constant is crucial because derivative operations eliminate constants, and integration seeks to include them back.
• In our solution, when we substitute \( u = e^x \) back into \( \sin(u) + C \), the answer becomes \( \sin(e^x) + C \).

\(\)Failing to include \( C \) implies the solution only represents one particular antiderivative, not the infinite possible family. Remembering \( C \) ensures completeness and correctness in indefinite integral notation.