Question

In the following exercises, find each indefinite integral by using appropriate substitutions. $$ \int x^{2} e^{-x^{3}} d x $$

Short answer

The indefinite integral is \(-\frac{1}{3} e^{-x^3} + C\).

Step-by-step solution

Recognize the need for substitution

Identify the part of the integrand that makes it complex to integrate directly. Notice that the integrand contains the exponential function \(e^{-x^3}\). This suggests using a substitution to simplify the exponent's complexity.

Choose the substitution

Choose a substitution that will simplify the integration. Let \(u = -x^3\) since its derivative \(-3x^2\) will relate to \(x^2\) in the function. This will help in transforming the integral into a simpler form.

Compute the derivative and solve for dx

Differentiate the substitution equation \(u = -x^3\) to find \(du\). We have \(du = -3x^2 dx\). Solving for \(dx\), we get \(dx = \frac{du}{-3x^2}\).

Substitute in the integral

Replace \(x^2\), \(e^{-x^3}\), and \(dx\) in the integral using our substitution. Substitute: \(x^2 = -\frac{du}{3}\), \(e^{-x^3} = e^u\), \(dx = \frac{du}{-3x^2}\). Simplifying gives the integral \(\int -\frac{1}{3} e^u du\).

Solve the simplified integral

Integrate \(\int -\frac{1}{3} e^u du\). The solution is \(-\frac{1}{3} e^u + C\), where \(C\) is the constant of integration.

Substitute back to original variable

Replace \(u\) with \(-x^3\) to get the integral in terms of \(x\). This gives \(-\frac{1}{3} e^{-x^3} + C\).

Key concepts

Substitution Method

The substitution method is a useful technique for simplifying the process of finding indefinite integrals, especially when the integrand involves complex functions. In this instance, the function inside the integral, \(x^2 e^{-x^3}\), would be challenging to handle directly.
Steps for Substitution:

• Identify a Part of the Integrand: Look for a section of the equation that complicates direct integration. Here, \(e^{-x^3}\) was complex because of its exponent.
• Choose Substitution: Select a substitution that simplifies the problem. For our exercise, let \(u = -x^3\). The reason for this choice is the close relationship between its derivative \(-3x^2\), which can replace the \(x^2\) term in the integral.
• Derive \(dx\): Differentiate \(u = -x^3\) to find \(du = -3x^2 dx\). This step allows you to express \(dx\) in terms of \(du\), facilitating the substitution process.

After making these transformations, the complicated integral \(\int x^2 e^{-x^3} dx\) is simplified into a much easier form, \(\int -\frac{1}{3} e^u du\). This simplification is a great demonstration of the power of substitution.

Exponential Function Integration

Exponential function integration is a fundamental aspect of calculus. Integrating exponential functions often involves recognizing patterns and using substitution to simplify the function, as seen in our example with the exponential term \(e^{-x^3}\).
Steps for Integrating Exponential Functions:

• Simplify using Substitution: Substitution helps derive a simpler expression. In our case, after substituting \(u = -x^3\), the integral becomes \(\int e^u du\).
• Integrate the Simplified Form: The integral of \(e^u\) is straightforward, simply \(e^u\). When it’s multiplied by a constant, such as \(-\frac{1}{3}\), it involves integrating \(-\frac{1}{3} e^u\). This gives \(-\frac{1}{3} e^u + C\), where \(C\) is the constant of integration.

This easy integration of exponential functions illustrates how initial transformations pave the way for more straightforward solutions. It reveals that once in a standard form, integrating exponential functions can be direct and efficient.

Calculus Problem Solving

Solving calculus problems effectively involves understanding various strategies and methods, such as the substitution method and recognizing function types, like exponential functions. Yes, this might initially seem overwhelming, but with practice and methodical steps, it becomes manageable.
General Tips for Solving Calculus Problems:

• Break Down the Problem: Divide the problem into smaller, more manageable steps. First, identify complex parts of the integrand.
• Choose the Right Method: Decide whether substitution, by-parts, or another method is more suitable. The correct approach can simplify the problem significantly.
• Follow Through with Calculations: After substitution, perform the calculations to transform and then integrate the simplified equation.
• Check Solutions: Always reframe back to the original variable and verify your integration step to avoid mistakes.
• Practice: Regular problem-solving practices cement understanding and application skills of learned concepts.

Applying these processes equips students to navigate the broad array of calculus problems systematically and confidently.