Problem 45 - Essential Calculus

Question

Find the indefinite integral. $$ \int e^{x}\left(e^{x}+1\right)^{2} d x $$

Step-by-Step Solution

Answer

\(\int e^{x}(e^{x}+1)^{2}\, dx = \frac{1}{3} (e^{x}+1)^3 + C \)

Step 1: Substitution

Set \( u=e^{x}+1\). Then differentiate \( u\) to find \( du\), which is \( du=e^{x} dx\). Now, the integral looks like: \[ \int u^{2} du \]

Step 2: Compute Integral

Now, compute the integral of a power function. The integral is obtained using the power rule of integration, which is given by: \[\int u^{n} du = \frac{1}{n+1} u^{n+1}\]Therefore, the integral becomes:\[\frac{1}{3} u^{3} + C\]where C is the constant of integration.

Step 3: Back-substitution

Finally, to get the answer in terms of \( x\), substitute back \( u = e^{x}+1\), yielding:\[\frac{1}{3} (e^{x}+1)^3 + C \]

Key Concepts

Substitution Method

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.

Power Rule of Integration

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.

Back-Substitution

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.