Solving Integrals with Limits: A Step by Step Guide

In summary, the conversation revolved around solving an integral using the substitution method. The person attempted to use the substitution x=ln(u), but ran into difficulty when trying to integrate cos(u)/u. They asked for help and were reminded that the limit should be taken as n->infinity instead of x->infinity. It was also noted that cos(e^x) is a bounded function and the limit of the integral can be found without actually solving it. The person then used the Mean Value Theorem to find the limit as n->infinity and concluded that it would be zero.
  • #1
rey242
41
0
Limit with integrel

Homework Statement


equationrender.png


Homework Equations





The Attempt at a Solution


I tried to take
x=ln(u)
dx=du/u
and solve the integral but I keep getting stuck at the int(cos(u)/u.)
Can anyone help me out here?
 
Physics news on Phys.org
  • #2


You can't take the limit x->infinity. x is a dummy integration variable. You must mean n->infinity. Right?
 
  • #3


I meant N.
Sorry, its a force of habit.
 
  • #4


Ok, then as n->infinity then 1/n -> 0. You don't actually have to do the integral to know what the limit is, because cos(e^x) is bounded. What's the limit of the integral of a bounded function between 0 and 1/n as n->infinity?
 
  • #5


so the integral would be zero as n approaches inf. since the limits of integration go from zero to zero, right?
 
  • #6


No, it's not just the limits of integration, you have to think about how the function you are integrating behaves. -1<=cos(e^x)<=1. Agree? So what limits can you make for the integral from 0 to 1/n of cos(e^x)? What happens as n->infinity?
 
  • #7


I'd try the MVT.
 

1. How do I determine the limits of integration?

The limits of integration are typically provided in the integral equation or can be determined by the context of the problem. They represent the range of values over which the integral will be evaluated.

2. What is the process for solving integrals with limits?

The general steps for solving integrals with limits are: 1) Identify the limits of integration, 2) Rewriting the integral in the proper format, 3) Evaluating the integral using appropriate techniques, and 4) Checking the answer for correctness.

3. What are some common techniques for solving integrals with limits?

Some common techniques for solving integrals with limits include substitution, integration by parts, and partial fraction decomposition. It is important to choose the appropriate technique based on the form of the integral and the limits involved.

4. How do I know if my answer is correct when solving integrals with limits?

You can check your answer by taking the derivative of your solution. If the derivative is equal to the original integrand, then your solution is correct. Additionally, you can use a graphing calculator or online calculator to verify your answer.

5. Can I use a calculator to solve integrals with limits?

Yes, you can use a graphing calculator or an online calculator to evaluate integrals. However, it is important to understand the steps and techniques involved in solving integrals with limits in order to use the calculator effectively.

Similar threads

Differentiate the given integral
    • Calculus and Beyond Homework Help
Replies
15
Views
787
Jacobian: how to change limits of integration?
    • Calculus and Beyond Homework Help
Replies
16
Views
1K
Solve the problem involving the given double integral
    • Calculus and Beyond Homework Help
Replies
1
Views
493
Integrals that keep me up at night
    • Calculus and Beyond Homework Help
Replies
22
Views
1K
How Do You Solve the Differential Equation dy/dx = 1 - y^2?
    • Calculus and Beyond Homework Help
Replies
12
Views
1K
Integral using substitution x = -u
    • Calculus and Beyond Homework Help
Replies
12
Views
1K
Solve the given problem that involves integration
    • Calculus and Beyond Homework Help
Replies
4
Views
740
Find the value of the definite integral
    • Calculus and Beyond Homework Help
Replies
8
Views
763
Solving the Integral ∫dx/(1-x)
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
Solve ##\int\frac{e^{-x}}{x^2}dx## and ##\int \frac{e^{-x}}{x}dx##
    • Calculus and Beyond Homework Help
Replies
7
Views
706

Hot Threads

Recent Insights

Back
Top