Maximum value of this integral

[Kqwert]

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?

 

[FactChecker]

To find local maximums / minimums of a function, look for the zeros of the derivative.

CORRECTION: This post is wrong. I misunderstood and was thinking about maximizing the range of the integral with a fixed value of a.

 

[Kqwert]

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.

 

[FactChecker]

On homework problems, I can only give hints to guide you. You must show what the derivative is and show some work to determine where the zeros are.
CORRECTION: This post is wrong. I misunderstood and was thinking about maximizing the range of the integral with a fixed value of a.

 

[SammyS]

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?

The method you use probably should be guided by topics you are currently studying.

It may be more rasonable to do this by evaluating the integral and then finding the maximum of the resulting function of a.

 

[FactChecker]

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.

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]

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?

 

[SammyS]

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?

Yes, that is correct.