/*! 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 19 evaluate the integral, and check... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 19

evaluate the integral, and check your answer by differentiating. $$\int\left[\frac{2}{x}+3 e^{x}\right] d x$$

Short Answer

Expert verified
The integral is \( 2\ln |x| + 3e^x + C \).

Step by step solution

01

Identify the Integral Components

The integral \( \int \left[\frac{2}{x}+3 e^{x}\right] dx \) consists of two separate functions: \( \frac{2}{x} \) and \( 3e^x \). We'll address each term individually.
02

Integrate the First Term

The first term is \( \frac{2}{x} \). The integral of \( \frac{1}{x} \) is \( \ln |x| \). Therefore, the integral of \( \frac{2}{x} \) is \( 2 \ln |x| \).
03

Integrate the Second Term

The second term is \( 3e^x \). The integral of \( e^x \) is \( e^x \). Therefore, the integral of \( 3e^x \) is \( 3e^x \).
04

Combine the Results

Combine the integrals from Step 2 and Step 3: \( \int \left[\frac{2}{x} + 3e^x \right] dx = 2\ln |x| + 3e^x + C \), where \( C \) is the constant of integration.
05

Differentiate to Verify

Differentiate the resultant expression: \( F(x) = 2\ln |x| + 3e^x + C \). The derivative of \( 2\ln |x| \) is \( \frac{2}{x} \) and the derivative of \( 3e^x \) is \( 3e^x \). Therefore, \( F'(x) = \frac{2}{x} + 3e^x \), which matches the original integrand.

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.

Differentiation
Differentiation is a fundamental concept in calculus used to determine the rate at which a function changes at any given point. In simpler terms, it tells us how a function's values are changing instantaneously.
Think of it as finding the slope of a curve at a particular point. This not only provides insights into the behavior of a function but also allows us to find tangents and optimize performance in various applications.
  • Differentiation helps in understanding motion and rates of change, like velocity.
  • It is crucial in determining maxima and minima for function optimization problems.
  • It provides a way to check if an integral solution is correct by reversing the process of integration.
If you've integrated a function to find an antiderivative, differentiating the result should bring you back to the original function, verifying accuracy.
Natural Logarithm
The natural logarithm, denoted as \(\ln(x)\), is a logarithm to the base "e", where "e" is an irrational and transcendental number approximately equal to 2.71828. It is a fundamental mathematical constant, frequently popping up in calculus, particularly when dealing with rates of growth or decay.
  • Natural logarithms transform multiplicative processes into additive ones, simplifying complex calculations.
  • They are instrumental in solving equations involving exponential decay or growth models.
  • The integral of \(\frac{1}{x}\) is a prime example of where the natural logarithm naturally appears in calculus: \(\int \frac{1}{x} \, dx = \ln |x| + C\).
When you integrate expressions like \(\frac{2}{x}\), the logarithmic function emerges as \(2 \ln |x|\), illustrating the profound connection between logarithms and rates of change.
Exponential Function
Exponential functions are equations where the variable appears in the exponent, such as \(e^x\). These functions are pivotal in modeling continuous growth processes, like population growth, radioactivity, or investment growth.
The base "e" provides a unique and natural growth pattern, frequently used in economic and scientific computations.
  • The exponential function \(e^x\) remains unaffected by differentiation, meaning the derivative of \(e^x\) is always \(e^x\).
  • Exponential growth models can be simplified with exponential functions, making them practical in real-world applications.
  • Integrating \(3e^x\) yields \(3e^x\) once more, capturing the growth rate directly in its function.
Understanding how exponential functions integrate and differentiate provides powerful tools in predicting and analyzing complex systems. It underscores the cyclical beauty of calculus, where functions and their rates of change are intricately linked.

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

(a) Make a conjecture about the value of the limit $$\lim _{k \rightarrow 0} \int_{1}^{x} t^{k-1} d t \quad(x>0)$$ (b) Check your conjecture by evaluating the integral, and then using L'H么pital's rule to find the limit.

Evaluate the definite integral by expressing it in terms of \(u\) and evaluating the resulting integral using a formula from geometry. $$\int_{\pi / 3}^{\pi / 2} \sin \theta \sqrt{1-4 \cos ^{2} \theta} d \theta ; u=2 \cos \theta$$

Let \(I=\int_{-1}^{1} \frac{1}{1+x^{2}} d x .\) Show that the substitution \(x=1 / u\) results in $$ I=-\int_{-1}^{1} \frac{1}{1+u^{2}} d u=-I $$ so \(2 I=0,\) which implies that \(I=0 .\) However, this is impossible since the integrand of the given integral is positive over the interval of integration. Where is the error?

Find the average value of the function over the given interval. $$f(x)=3 x ;[1,3]$$

Evaluate the integrals by any method. $$\int_{\pi^{2}}^{4 \pi^{2}} \frac{1}{\sqrt{x}} \sin \sqrt{x} d x$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

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.