/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 Find the integrals Check your an... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 10

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

Short Answer

Expert verified
The integral is \( -\frac{1}{2} e^{-x^2} + C \).

Step by step solution

01

Identify the Method of Integration

For the integral \( \int x e^{-x^{2}} \, dx \), we notice that the integrand contains a product of a polynomial \( x \) and an exponential function \( e^{-x^2} \). This suggests the use of substitution might be effective.
02

Choose an Appropriate Substitution

Let \( u = -x^2 \). Then, \( du = -2x \, dx \), or equivalently \( x \, dx = -\frac{1}{2} du \). This will simplify the integral.
03

Substitute and Simplify the Integral

Substitute into the integral: \[\int x e^{-x^2} \, dx = \int e^u \left(-\frac{1}{2} \right) du = -\frac{1}{2} \int e^u \, du\]Simplify this to obtain:\[-\frac{1}{2} e^u + C\]
04

Back-Substitute to Original Variable

Return to the original variable:\[-\frac{1}{2} e^{-x^2} + C\]This is the antiderivative of \( \int x e^{-x^2} \, dx \).
05

Differentiate to Check Your Work

Now differentiate the found function to see if it matches the original integrand:\[d\left(-\frac{1}{2} e^{-x^2} + C\right)/dx = \left(-\frac{1}{2}\right)(-2x)e^{-x^2} = x e^{-x^2}\]This confirms that our integration is correct, as taking the derivative gives us the original function.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution Method
The substitution method is a powerful technique when it comes to integration, especially helpful for integrals that involve composite functions. The key idea is to change the variable of integration to something simpler:
  • We find a substitution that simplifies the integrand. Often, this involves setting a part of the integrand equal to a new variable, like choosing a complex expression to become a single variable.
  • In our example, we set \( u = -x^2 \). This cleverly transforms the variable and makes the exponential function easier to handle, as it removes the complexity of \(-x^2\).
  • After substituting, remember to adjust the differential as well. We found \( du = -2x \, dx \), so \( x \, dx = -\frac{1}{2} du \).
This change turns the integral into a much simpler form. Once solved, the substitution is reversed (back-substitution) to reach the final result in terms of the original variable.
Exponential Functions
Exponential functions are key to understanding many natural phenomena, described by expressions like \( e^{x} \), where \( e \) is Euler's number, approximately equal to 2.718. A notable aspect is their unique property:
  • When differentiating or integrating exponential functions like \( e^{u} \), the operation doesn鈥檛 alter the function鈥檚 form.
  • For integration in our example, \( e^{-x^2} \) changed to \( e^{u} \) due to substitution. This allowed us to proceed with resolving \( \int e^{u} du \).
  • As we saw, integrating gives us \( e^{u} + C \), where \( C \) is a constant of integration.
This behavior of exponential functions greatly simplifies the process of integration and differentiation, as they typically remain in the same form.
Differentiation
Differentiation is the counterpart to integration, where we determine the rate of change of a function. It plays a crucial role in verifying the correctness of an integral:
  • In our process, after finding an antiderivative, we differentiated \( -\frac{1}{2} e^{-x^2} + C \) to ensure it matched \( x e^{-x^2} \), the original integrand.
  • The differentiation involved using the chain rule. Since \( e^{-x^2} \) had an inner function \(-x^2\), its derivative gave \(-2x e^{-x^2} \).
  • Combining this with the constant multiplier \(-\frac{1}{2}\) results in \( x e^{-x^2} \), affirming our integration was conducted correctly.
This step is crucial in confirming the integration process, ensuring the derived function truly serves as the solution to the original integral.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the exact area of the regions.Bounded by \(y=x^{3} / \sqrt{4-x^{2}}, y=0, x=0, x=\sqrt{2}\).

Decide whether the statements are true or false. Give an explanation for your answer. $$\int f^{\prime}(x) \cos (f(x)) d x=\sin (f(x))+C$$.

Show the two integrals are equal using a substitution. $$\int_{0}^{\pi} x \cos (\pi-x) d x=\int_{0}^{x}(\pi-t) \cos t d t$$

Integrate: (a) \(\int \frac{1}{\sqrt{x}} d x\) (b) \(\int \frac{1}{\sqrt{x+1}} d x\) (c) \(\int \frac{1}{\sqrt{x}+1} d x\)

Let \(f(t)\) be the rate of flow, in cubic meters per hour, of a flooding river at time \(t\) in hours. Give an integral for the total flow of the river (a) Over the 3 -day period \(0 \leq t \leq 72\) (b) In terms of time \(T\) in days over the same 3 -day period.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.