# Maximum value of this integral

• Kqwert
In summary, the method used to find the optimal value of a for the given integral is to calculate the integral and take the derivative of the resulting function with respect to a. The maximum value of the function is found at a = 1, making 1 the optimal value for the integral to have the highest possible value.
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?

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.

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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.

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.

Last edited:
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?
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
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.
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.

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?

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?
Yes, that is correct.

## 1. What is the maximum value of an integral?

The maximum value of an integral is the largest possible numerical value that can be obtained by evaluating the integral over a given interval.

## 2. How is the maximum value of an integral determined?

The maximum value of an integral can be determined by using mathematical techniques such as the first and second derivative tests, or by graphing the function and finding the highest point on the curve within the given interval.

## 3. Can an integral have multiple maximum values?

No, an integral can only have one maximum value within a given interval.

## 4. Does the maximum value of an integral change if the interval is changed?

Yes, the maximum value of an integral may change if the interval is changed. It is important to evaluate the integral over the entire interval of interest to determine the true maximum value.

## 5. Is the maximum value of an integral always positive?

No, the maximum value of an integral can be positive, negative, or zero. It depends on the function being integrated and the interval of integration.

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