The discussion revolves around finding a positive value of \( a \) that maximizes the integral of the function \( e^{-ax} \cos x \) from 0 to infinity. Participants are exploring the relationship between the parameter \( a \) and the value of the integral.
The conversation includes attempts to clarify the approach to maximizing the integral, with some participants providing hints about evaluating the integral and finding its maximum with respect to \( a \). There is recognition of misunderstandings regarding the problem setup, and some participants express uncertainty about the implications of their calculations.
Participants mention constraints related to homework guidelines, emphasizing the need to show work and derive results rather than simply stating conclusions. There is also a focus on the necessity of understanding the relationship between the integral and its parameters.
The method you use probably should be guided by topics you are currently studying.Kqwert said:Homework Statement
Find a > 0 so the integral
int(exp(-ax)*cosx)dx from 0 to inf get as high value as possible.
The Attempt at a Solution
My way of solving this is to plot the integrand, i.e. exp(-ax)*cosx and check for different values of a. The larger a is, the smaller the area under the curve from 0 to inf gets, i.e. a should be as small as possible.
Is this the correct way of doing it?
Sorry, I misunderstood the question. You should calculate the integral and determine the value of a that gives a minimum. To do that, it may be necessary to take the derivative of the integral with respect to a and determine when it is 0.Kqwert said:But how do I do that when a is unknown? I know how to derivate the function, but not what I know from doing that.
Yes, that is correct.Kqwert said:So I calculated the integral, which resulted in a/(a^2+1). This has a maximum value for a = 1, i.e. a should be 1. Is this correct?