/*! 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 50 In Problems 49-60, use either su... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 50

In Problems 49-60, use either substitution or integration by parts to evaluate each integral. $$ \int x e^{-2 x^{2}} d x $$

Short Answer

Expert verified
The integral \( \int x e^{-2x^2} dx = -\frac{1}{4} e^{-2x^2} + C \).

Step by step solution

01

Choose the Integration Method

Given the integral \( \int x e^{-2x^2} dx \), we will use substitution as it is suitable for integrals involving polynomial expressions and their derivatives. Integration by parts is not ideal here because we don't have a clear product of functions to use.
02

Substitution Setup

We aim to simplify \( e^{-2x^2} \). Let's choose the substitution \( u = -2x^2 \). This gives us \( du = -4x \, dx \). Consequently, solving for \( x \, dx \) gives \( x \, dx = -\frac{1}{4} du \).
03

Substitution Application

Substitute \( u = -2x^2 \) and \( x \, dx = -\frac{1}{4} du \) into the original integral:\[ \int x e^{-2x^2} \, dx = \int e^u \left(-\frac{1}{4} \, du\right) \]This simplifies to:\[ -\frac{1}{4} \int e^u \, du \]
04

Integrate with Respect to u

The integral \( \int e^u \, du \) is simply \( e^u \). Thus, the integration becomes:\[ -\frac{1}{4} e^u + C \]Where \( C \) is the constant of integration.
05

Substitute Back to x

Replace \( u \) with \( -2x^2 \) to express the solution in terms of \( x \):\[ -\frac{1}{4} e^{-2x^2} + C \]

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 in calculus is a powerful tool for simplifying integrals that are otherwise difficult to solve. The key idea is to replace a complicated part of the integral with a single variable, usually denoted as \( u \). This method involves three main steps:
  • **Choosing a Substitution:** Identify a part of the integral that, when substituted with \( u \), makes the integral easier to evaluate. Common choices include expressions that appear frequently and their derivatives.
  • **Calculate \( du \):** Once you choose your substitution, differentiate it to find \( du \). This step allows you to replace \( dx \) or any other derivatives in the integral.
  • **Rewrite and Integrate:** Substitute all parts of the integral accordingly and then perform the integration with respect to \( u \). Solve the simplified integral and don't forget to convert back to the original variable, \( x \).
The substitution method can be viewed as "undoing" the chain rule of differentiation. In our exercise, deciding to use \( u = -2x^2 \) transforms the exponential integral into a simpler form involving \( e^u \). This is particularly useful when a derivative, such as \( du = -4x \, dx \), aligns with other parts of the integral, making substitution straightforward.
Integration by Parts
Integration by parts is another technique for solving integrals, based on the product rule for differentiation. The formula it uses is:\[\int u \, dv = uv - \int v \, du\]This method is handy when the integral is a product of two functions. The steps involve:
  • **Identify Parts:** Choose which part of the integral will be \( u \) and which will be \( dv \). A common strategy is using the LIATE rule鈥擫ogarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential.
  • **Differentiate and Integrate:** Differentiate \( u \) to find \( du \) and integrate \( dv \) to find \( v \).
  • **Apply the Formula:** Calculate \( uv - \int v \, du \) to solve the integral.
Different from substitution, integration by parts targets integrals that naturally separate into products. It's less applicable to our example integral because the function \( e^{-2x^2} \) with an \( x \) factor doesn鈥檛 split nicely into independent terms. Hence, the substitution method offers a simpler route for this integral.
Indefinite Integral
An indefinite integral, often called an antiderivative, represents a family of functions whose derivative gives the original function. It's written without limits of integration, resulting in a general form that includes a constant, \( C \). Here's how it works:
  • **Result of Integration:** Calculating an indefinite integral results in a general antiderivative plus an arbitrary constant, \( C \). This accounts for the fact that the derivative of a constant is zero, meaning the original function could have been any constant plus another function.
  • **Notation:** Represented as \( \int f(x) \ dx \), it's signified by a solitary integral sign and involves evaluating the antiderivative of \( f(x) \).
  • **Purpose:** Indefinite integrals are used to determine the original functions from their derivatives, helping in solving differential equations and problems within physical sciences.
In the exercise, the indefinite integral \( \int x e^{-2x^2} \ dx \) results in \(-\frac{1}{4} e^{-2x^2} + C \). The constant \( C \) is crucial, reminding us of the infinite potential horizontal shifts along the \( y \)-axis for this family of solutions.

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

Use the midpoint rule to approximate each integral with the specified value of \(n\). $$ \int_{0}^{1} \sin \left(x^{2}\right) d x, n=4 $$

Find a comparison function for each integrand and determine whether the integral is convergent. $$ \int_{1}^{\infty} \frac{1}{\sqrt{1+x}} d x $$

Use the midpoint rule to approximate each integral with the specified value of \(n .\) Compare your approximation with the exact value. $$ \int_{0}^{1}\left(e^{2 x}-1\right) d x, n=4 $$

Show that $$ x^{4} \approx a^{4}+4 a^{3}(x-a) $$ for \(x\) close to \(a\).

Compute the Taylor polynomial of degree \(n\) about \(x=0\) for each function and compare the value of the function at the indicated point with the value of the corresponding Taylor polynomial. $$ f(x)=e^{2 x}, n=4, x=0.3 $$
See all solutions

Recommended explanations on Biology Textbooks

Substance Exchange

Read Explanation

Cell Cycle

Read Explanation

Biological Molecules

Read Explanation

Biological Organisms

Read Explanation

Control of Gene Expression

Read Explanation

Plant Biology

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.