Question

Find the integral by using the simplest method. Not all problems require integration by parts. $$ \int x e^{-x^{2}} d x $$

Short answer

\( -\frac{1}{2} e^{-x^2} + C \).

Step-by-step solution

Identify the Integration Method

The given integral is \( \int x e^{-x^2} \, dx \). Observing the integrand, we see that the derivative of \( x^2 \) is \( 2x \). This suggests that substitution might be a suitable method for this integral.

Choose the Substitution Variable

Let \( u = -x^2 \). Then, differentiating both sides with respect to \( x \), we obtain \( du = -2x \, dx \). We can solve for \( x \, dx \) to get \( x \, dx = -\frac{1}{2} du \).

Substitute and Rewrite the Integral

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

Integrate with Respect to \( u \)

The integral \( \int -\frac{1}{2} e^u \, du \) can be calculated directly. It is \( -\frac{1}{2} \int e^u \, du = -\frac{1}{2} e^u + C \), where \( C \) is the constant of integration.

Substitute Back to Original Variable

Replace \( u \, \) back with \( -x^2 \) in the integrated result: \[ -\frac{1}{2} e^u + C = -\frac{1}{2} e^{-x^2} + C. \] This represents the integral in terms of the original variable \( x \).

Key concepts

Substitution Method

The substitution method is a fundamental integration technique that simplifies a complicated integral by changing variables. Essentially, the goal is to transform the integral into a more recognizable form that is easier to solve. In this exercise, the given integral is \( \int x e^{-x^2} \, dx \). By observing the integrand, we notice the presence of \( x \) and \( e^{-x^2} \), which hints that substitution might help.

Choosing the Right Substitution:
- We set \( u = -x^2 \), a common choice when dealing with function compositions like \( e^{-x^2} \).
- The differential \( du = -2x \, dx \) allows us to express \( x \, dx \) in terms of \( du \), specifically, \( x \, dx = -\frac{1}{2} du \).

Substituting into the integral, it becomes \( \int -\frac{1}{2} e^u \, du \), which is simpler to integrate. With practice, the substitution method becomes a powerful tool that reveals the solution to integrals that seem complex initially.

Exponential Functions

Exponential functions are a crucial component of calculus, characterized by the function \( e^x \). They have unique properties that simplify differentiation and integration processes. In this exercise, the focus is not just on \( e^x \) but on \( e^{-x^2} \), which is pivotal to solving the integral. Understanding the behavior of exponential functions helps in identifying the right integration approach.

Key Properties of Exponential Functions:
- The derivative of \( e^u \) with respect to \( u \) is \( e^u \) itself.
- This property makes integrating \( e^u \) straightforward, as it mirrors its own form.

In this exercise, after using substitution, the integral resembles \( \int e^u \, du \). This is directly integrable as \( e^u + C \), illustrating how exponential functions maintain simplicity and elegance even through transformations.

Indefinite Integrals

Indefinite integrals, represented as \( \int f(x) \, dx \), signify antiderivatives of functions. They are foundational in calculus, revealing functions whose derivatives match the integrand. The integral in this problem \( \int x e^{-x^2} \, dx \) exemplifies an indefinite integral, requiring a combination of techniques to solve.

Understanding Indefinite Integrals:
- Unlike definite integrals, indefinite integrals indicate a general expression with a constant of integration, \( C \).
- Solving involves techniques like substitution, which transforms the integral into a simpler form as shown in this exercise.

After substitution and simplifying the integral to \( \int -\frac{1}{2} e^u \, du \), the result \(-\frac{1}{2} e^{u} + C\) must be expressed in terms of the original variables for full comprehension. Consequently, substituting back, we conclude with \(-\frac{1}{2} e^{-x^2} + C\), reaffirming the indefinite nature with the inclusion of \( C \).