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The substitution method is a powerful technique used in calculus to evaluate integrals. It's particularly useful when dealing with integrals that include a composite function—functions of functions. By effectively transforming the variables in an integral, this method simplifies the integration process.To begin using the substitution method, identify a substitution that will simplify the integral. This typically involves recognizing a portion of the integrand (the function being integrated) which can be replaced by a new variable, commonly denoted as "u."

• Choose a function within the integral to set as your substitution. In our original exercise, the substitution is set as: \( u = 1 - 3x^2 \).
• Calculate the derivative of this new variable with respect to the original variable, \( x \), to express \( dx \) in terms of \( du \). This aids in transforming the entire integral into terms of \( u \).

When you substitute \( u \) into the integral, the resulting expression should be more straightforward to integrate. The goal is to replace all occurrences of the original variable, such as \( x \), making the integral solely in terms of \( u \). This often results in a simpler function that is easier to integrate.

After integrating with the substitution method, the resulting integral will be in terms of the new variable \( u \). At this point, we need to express our answer once again in terms of \( x \), which is known as back-substitution.Back-substitute by reversing the substitution made at the beginning:

• Take the antiderivative found after integration in terms of \( u \), in our case \( -\frac{1}{6} e^{u} + C \), and replace \( u \) with the expression that was originally substituted: \( 1 - 3x^2 \).
• The final expression after back-substitution for our example is \( -\frac{1}{6} e^{1-3x^2} + C \).

This step is essential because the initial problem was in terms of \( x \), and we need the final solution in the same variable for it to be meaningful and properly solve the original problem.

In the process of indefinite integration, as opposed to definite integration, we introduce a constant known as the integration constant, denoted by \( C \). This constant accounts for the family of antiderivatives.When you evaluate an indefinite integral, such as in our example, the result includes \( C \). Why is this necessary?

• Indefinite integrals represent a family of functions, all of which differ by a constant. Thus, without \( C \), there would be no way to encapsulate all potential vertical shifts of the function.
• The constant \( C \) ensures that every possible antiderivative of the function is considered, since taking the derivative of \( C \) gives 0, aligning with the definition of an antiderivative.

The presence of \( C \) in the final result reflects the inherent ambiguity in antiderivatives until a specific boundary condition or additional information is provided.