Problem 12 - Applied Calculus

Question

Find the integrals .Check your answers by differentiation. $$\int x e^{-x^{2}} d x$$

Step-by-Step Solution

Answer

The integral is \(-\frac{1}{2} e^{-x^2} + C\). Differentiating confirms it matches the original integrand.

Step 1: Choose a Method

Since the integral involves both polynomial and exponential terms, we will use integration by substitution. Identify a substitution that simplifies the integral.

Step 2: Define the Substitution

Let us choose the substitution \( u = -x^2 \). Then differentiate both sides, which gives \( du = -2x \, dx \) or \( x \, dx = -\frac{1}{2} du \). This simplifies our integral.

Step 3: Transform the Integral

Substitute \( u = -x^2 \) and \( x \, dx = -\frac{1}{2} du \) into the integral: \[ \int x e^{-x^2} \, dx = \int e^u \left( -\frac{1}{2} \right) \, du = -\frac{1}{2} \int e^u \, du \]

Step 4: Integrate

Now integrate with respect to \( u \): \[ -\frac{1}{2} \int e^u \, du = -\frac{1}{2} e^u + C \] where \( C \) is the constant of integration.

Step 5: Substitute Back x

Change \( u \) back to \( x \) using \( u = -x^2 \): \[ -\frac{1}{2} e^u + C = -\frac{1}{2} e^{-x^2} + C \]

Step 6: Differentiate to Check

Differentiate \( -\frac{1}{2} e^{-x^2} + C \) with respect to \( x \) to check the integration result: \[\frac{d}{dx} \left( -\frac{1}{2} e^{-x^2} \right) = -\frac{1}{2} \left( e^{-x^2} \right)(-2x) = x e^{-x^2}\]Thus, the derivative matches the original integrand \( x e^{-x^2} \).

Key Concepts

Integration by Substitution

Integration by substitution is a handy technique when dealing with complex integrals, especially when polynomial and exponential terms are involved. It transforms a difficult integral into a simpler form by substituting a part of the integral with a new variable.

This method is similar to the chain rule in differentiation but applied in the reverse order. Here’s how you can use this technique effectively:

• Identify which part of the integral makes it complex and choose a substitution that simplifies it.
• Differentiate the substitution to find the value of differential elements in terms of the new variable.
• Replace the original integral with the substituted variables and integrals.

In the example, we chose the substitution \( u = -x^2 \), resulting in a simpler form \( \int e^u \, du \) after replacing the necessary elements. This simplification makes it easier to integrate, ultimately leading us to solve the problem efficiently.

Polynomial Terms

Polynomial terms are expressions consisting of variables raised to constant, typically non-negative integer powers. In the context of integration, being able to identify and handle these terms is crucial.

In our example, \( x \) is the polynomial term, making the initial integral \( x e^{-x^2} \, dx \) a product of a polynomial and an exponential function. It complicates direct integration, hence necessitating sophisticated techniques like integration by substitution.

• Keep an eye out for simple polynomial terms, as they often give clues for substitution methods, simplifying integrals.
• Derivative rules applied beforehand may aid in understanding potential substitutions.
• Recognize polynomial behavior when substituted forms are used in the integration process, simplifying to constants.

Understanding polynomial terms is key, as they form a foundational part of various integration strategies, turning daunting integrals into manageable tasks.

Exponential Functions

Exponential functions are critical in calculus due to their unique properties under differentiation and integration. These functions generally take the form \( e^x \), where \( e \) is the base of the natural logarithm.

In the given problem, we encountered an exponential function \( e^{-x^2} \). Generally, integrating functions with exponentials can be challenging without a proper method:

• Recognize that many exponential integrals can be transformed into simpler forms, particularly by using substitutions.
• Understand how exponential functions behave under integration and differentiation to simplify or solve the problems.
• In our example, converting \( x e^{-x^2} \, dx \) into \( e^u \) through substitution made the problem tractable and integration feasible.

Exponential functions play a pivotal role in calculus due to their continuous growth or decay models, making it essential to grasp their behavior and handling within integrals.

Differentiation

Differentiation is the process of finding the derivative of a function, illustrating how a function changes as its inputs change. In the context of integration, differentiation underpins techniques to verify solutions.

For example, once we solved the integral, we checked our work by differentiating \( -\frac{1}{2} e^{-x^2} + C \) to ensure we obtained our original integrand.

• Verify integration results by differentiating the solution; the derivative should match the initial integrand.
• Utilize differentiation rules such as the product, chain, and power rules, which frequently appear in integration alongside substitutions.
• Recognize differentiation simplifies confirming solutions by linking integration processes back to initial problem statements.

Differentiation not only helps confirm integration results but enriches one's understanding, offering alternative perspectives on handling calculus problems effectively.