/*! 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 9 Evaluate the indefinite integral... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 9

Evaluate the indefinite integral by making the given substitution. $$ \int e^{2 x+3} d x, \text { with } u=2 x+3 $$

Short Answer

Expert verified
The indefinite integral is \( \frac{1}{2} e^{2x+3} + C \).

Step by step solution

01

Identify the Substitution

In this problem, the substitution given is \( u = 2x + 3 \). This means we need to replace \( 2x + 3 \) with \( u \) in the integral.
02

Differentiate the Substitution

Differentiate \( u = 2x + 3 \) with respect to \( x \) to find \( du \). The derivative is \( \frac{du}{dx} = 2 \).
03

Solve for dx

Rearrange the derivative for \( dx \) as follows:\[ dx = \frac{du}{2} \]
04

Substitute in the Integral

Substitute \( u \) and \( dx \) into the integral. It turns from\[ \int e^{2x+3} \, dx \] to \[ \int e^{u} \, \frac{du}{2} \] or \[ \frac{1}{2} \int e^{u} \, du \] .
05

Integrate with respect to u

Integrate \( \frac{1}{2} \int e^u \, du \). The result is \[ \frac{1}{2} e^u + C \] where \( C \) is the constant of integration.
06

Back-Substitute for u

Replace \( u \) with the original expression \( 2x + 3 \). The solution becomes\[ \frac{1}{2} e^{2x+3} + C \].

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 is a powerful technique used to simplify the process of integration. It works by replacing a complex expression with a simpler variable, making the integral easier to solve. In the given exercise, the substitution is given as \( u = 2x + 3 \).
  • This method often involves identifying a part of the integrand (the function inside the integral) that can be replaced by a single variable, \( u \).
  • Once the substitution is chosen, you differentiate it to find \( du \). This step is essential as it helps in changing the integration variable from \( x \) to \( u \).
  • In our exercise, differentiating \( u = 2x + 3 \) gives \( \frac{du}{dx} = 2 \), leading to \( dx = \frac{du}{2} \).
This transformation is crucial because it simplifies the integral to a form that is easier to integrate.
Integration
Integration is the mathematical process of finding the integral of a function. In the case of indefinite integration, the result includes a constant, \( C \), representing an infinite set of possible solutions.
  • With the substitution step complete, our integral \( \int e^{2x+3} \, dx \) changes to \( \frac{1}{2} \int e^u \, du \).
  • The integrand \( e^u \) is a basic exponential function, which simplifies the integration.
Once you find the antiderivative, which is \( \int e^u \, du = e^u \), the integral becomes \( \frac{1}{2} e^u + C \). This step transforms a complex expression into a simple one.
Exponential Function
Exponential functions are a key part of calculus and often appear in integration problems. These functions take the form \( e^x \), where \( e \) is the base of natural logarithms.
  • They have unique properties, such as the derivative and integral of \( e^x \) both being \( e^x \), which greatly simplifies the process of calculus operations.
  • In our problem, after substituting and integrating, \( e^u \) becomes the focus, and its integral is \( e^u \).
After integrating, it is important to return to the original variable by substituting \( u \) back with \( 2x + 3 \). This step gives us the final solution: \( \frac{1}{2} e^{2x+3} + C \). Understanding the behavior of exponential functions is crucial for successfully solving such integrals.

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 integration by parts to evaluate the integrals. $$ \int x^{2} \ln x d x $$

Use substitution to evaluate the definite integrals. $$ \int_{0}^{2} \frac{2 x}{\left(4 x^{2}+3\right)^{1 / 3}} d x $$

Use either substitution or integration by parts to evaluate each integral. $$ \int \frac{1}{x^{2}+5} d x $$

Use substitution to evaluate the definite integrals. $$ \int_{0}^{\pi / 3} \sin x \cos x d x $$

First make an appropriate substitution and then use integration by parts to evaluate the indefinite integrals. $$ \int x^{5} e^{x^{2}} d x $$
See all solutions

Recommended explanations on Biology Textbooks

Reproduction

Read Explanation

Bioenergetics

Read Explanation

Control of Gene Expression

Read Explanation

Cellular Energetics

Read Explanation

Ecosystems

Read Explanation

Biological Organisms

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.