Problem 7 - College Algebra and Calculus: An Applied Approach

Question

Use the Exponential Rule to find the indefinite integral. $$ \int 5 x^{2} e^{x^{3}} d x $$

Step-by-Step Solution

Answer

\( 5/3 e^{x^3} + C \)

Step 1: Apply the method of substitution

In order to simplify the integral, the substitution method can be used. Let \( u = x^3 \). The derivative \( du = 3x^2 dx \) is similar to the function inside of our integral. However, there is an extra coefficient of 5 instead of 3. As such, we adjust our \( du \) to fit the given function better, so we can say \( du/3 = x^2 dx \) instead.

Step 2: Transform the integral using substitution

Interchange the variables in the integral according to the substitution \( u=x^3 \), and \( du/3=x^2 dx \), we get \( \int 5 e^u du/3 \). We can take the constants out of the integral \( 5/3 \int e^u du \).

Step 3: Find the integral

This is a much simpler integral to solve. The integral of \( e^u \) with respect to \( u \) is just \( e^u \). So, completing this calculation gives us \( 5/3 e^u + C \).

Step 4: Substitute back

Finally, substitute back \( u = x^3 \) into the answer from step 3 to get the original variable back. This results in the final answer: \( 5/3 e^{x^3} + C \).

Key Concepts

Exponential Rule

Understanding the exponential rule is pivotal when dealing with integrals involving exponentials. In general, if you're integrating a function of the form \( e^{g(x)} \), where \( g(x) \) is a differentiable function of \( x \), the integral will take the form \( \frac{e^{g(x)}}{g'(x)} + C \), considering that \( g'(x) \) is nonzero. However, this requires an adjustment if the exponent has an additional scalar multiplier or if it’s not a simple exponent.

For the given exercise problem, we're faced with integrating \( 5x^2e^{x^3} \). To apply the exponential rule effectively, you must identify a substitution that will transform the integral into a form where the exponential rule can be applied directly. This is commonly done by recognizing a potential substitution for \( u \) that parallels the exponent in the exponential function, leading to a much simpler integral of \( e^u \).

Substitution Method

The substitution method is a technique often employed when an integral contains a composite function. The goal is to simplify the composite function to a basic form that can be easily integrated. In the context of our exercise, we recognize that \( x^3 \) is the inside function, and by setting \( u = x^3 \), we transform the integral into a simpler form that contains \( e^u \).

It's important to find the derivative of our substitution, in this case, \( du = 3x^2 dx \). Since the integral has a \( 5x^2 dx \) component, we adjust by finding an equivalent form of \( du \), which yields \( \frac{du}{3} = x^2 dx \). Consequently, this enables us to rewrite the entire integral in terms of \( u \), facilitating the application of the exponential rule without the complication of the original variables.

Integration Techniques

Integration techniques are numerous, each suited to a particular type of function or integral form. Apart from the substitution method, other common techniques include integration by parts, partial fractions, and trigonometric substitution. Understanding when and how to apply these methods is a critical skill in calculus.

The integration process for this particular exercise hinges on a strategic substitution, but in more complex scenarios, a combination of techniques may be necessary. For instance, after substitution, if the resulting integral was still complex, you might further employ integration by parts or another appropriate technique to arrive at the solution.

Calculus Education

Calculus education strives to equip students with the conceptual understanding and technical skills necessary to tackle a wide array of problems involving change and motion. Engaging with exercises such as finding indefinite integrals serves to reinforce the understanding of fundamental calculus concepts such as the exponential rule and substitution method.

Particularly, providing step-by-step solutions and explanations helps students make connections between the abstract principles they learn and the concrete problems they solve. This exercise serves as an educational tool not only to practice integration but also to develop problem-solving strategies by identifying suitable techniques and understanding their implementation in different scenarios.