91Ó°ÊÓ

The substitution method is a popular technique in integration that simplifies finding indefinite integrals by transforming the original variable. In this situation, we first determine a suitable expression to substitute. Here, we let \( u = e^x + 1 \) to simplify the given integral.

By substituting, we transform the integral into a form that is easier to manage, often reducing it to a standard form that is straightforward to integrate. For our initial function \( \int e^x(e^x + 1)^2 \, dx \), substituting \( u = e^x + 1 \) allows us to express the differential \( dx \) in terms of \( du \), using the derivative \( du = e^x \, dx \).

This change effectively turns the integral into \( \int u^2 \, du \). This transformation is advantageous because it removes the variable \( x \) from the equation, simplifying the integration process. Remember, identifying the right substitution can often be the key to unlocking more complex integration problems.

The power rule of integration is a straightforward technique used to integrate power functions easily. It states that for any variable \( u \), with a constant \( n \), the integral of \( u^n \) is given by \( \int u^n \, du = \frac{1}{n+1} u^{n+1} + C \), with \( C \) as the integration constant.

In the context of our exercise, after applying substitution, we are left with the integral \( \int u^2 \, du \). The power rule comes directly into play here, where \( n = 2 \).

Applying the rule, we find the integral is \( \frac{1}{3} u^3 + C \). The simplicity of the power rule makes it an invaluable tool for dealing with polynomials, as it systematically reduces the power by integrating and offers a straightforward solution for finding antiderivatives.

Once the integration process is completed using the substitution method, the next step is back-substitution. This technique involves replacing the substitution variable back with the original expression to express the final solution in terms of the initial variable.

In our example, after integrating \( \int u^2 \, du \) to \( \frac{1}{3} u^3 + C \), we substitute \( u \) back with \( e^x + 1 \).

Performing this back-substitution, we get \( \frac{1}{3} (e^x + 1)^3 + C \). This step concludes the problem, providing the integral in its original variable \( x \), bringing full circle to our original task. Back-substitution is crucial as it ensures the solution is applicable to the original context of the problem, bridging the temporarily shifted variables back to their real-world application.