Solving Integrals with Limits: A Step by Step Guide

[rey242]

Limit with integrel

Homework Statement

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?

 

[Dick]

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

 

[rey242]

I meant N.
Sorry, its a force of habit.

 

[Dick]

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?

 

[rey242]

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

 

[Dick]

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?

 

[JG89]

I'd try the MVT.