What is an important step in the process of integration by substitution?

What is an important step in the process of integration by substitution?

• A. Selecting a New Variable
• B. Identifying Limits of Integration
• C. Performing Long Division
• D. Using Integration by Parts

Answer: (A)

In the integration by substitution method, selecting a new variable is indeed a crucial step. This process involves identifying an appropriate function within the integral that can simplify the overall computation. By choosing a new variable, usually denoted as \( u \), and defining it as a function of the original variable, one can transform the integral into a more manageable form. The primary goal of this substitution is to simplify the integrand, making it easier to integrate. For instance, when faced with an integral that contains a composite function, substituting \( u \) for the inner function allows one to reduce the complexity, often turning the integrand into a polynomial or a simpler expression that can be easily integrated. Other steps, such as identifying limits of integration, would be relevant in the context of definite integrals after substitution has been made, and would follow the selection of the new variable. The process does not typically involve long division or integration by parts, as these methods serve different purposes in integration. Therefore, selecting the new variable stands out as a foundational step in the substitution technique.

In the integration by substitution method, selecting a new variable is indeed a crucial step. This process involves identifying an appropriate function within the integral that can simplify the overall computation. By choosing a new variable, usually denoted as ( u ), and defining it as a function of the original variable, one can transform the integral into a more manageable form.

The primary goal of this substitution is to simplify the integrand, making it easier to integrate. For instance, when faced with an integral that contains a composite function, substituting ( u ) for the inner function allows one to reduce the complexity, often turning the integrand into a polynomial or a simpler expression that can be easily integrated.

Other steps, such as identifying limits of integration, would be relevant in the context of definite integrals after substitution has been made, and would follow the selection of the new variable. The process does not typically involve long division or integration by parts, as these methods serve different purposes in integration. Therefore, selecting the new variable stands out as a foundational step in the substitution technique.