# What is the Use of the Constant in the Integral?

• songoku
In summary, the conversation is about finding the function f(x) where f(x) is equal to the integral of cos t over t, with the starting point of integration at p=3 and the ending point at x. The conversation also discusses the importance of including the constant that is added to the integral. Additionally, there is a clarification about the derivative of f(x) being equal to cos x over x.
songoku
Homework Statement
Find f(x) where ##f(x)=\frac{\cos x}{x}## and f(3) = 4. State the answer in form of ##f(x)=\int_{t=p}^{t=q} (........)##
Relevant Equations
Fundamental Theorem of Calculus
This is my attempt:
$$f(x)=\int_{t=p}^{t=x} \frac{\cos t}{t} dt$$

But I am not sure what ##p## is and what the use of ##f(3)=4##

Thanks

You forgot about the constant that is added to the integral. If you start the integral at p=3, then you know that the integral part is 0 at x=3. So what constant is added to the integral?

songoku and Delta2
songoku said:
Homework Statement:: Find f(x) where ##f(x)=\frac{\cos x}{x}## and f(3) = 4. State the answer in form of ##f(x)=\int_{t=p}^{t=q} (...)##
Did you forget to add the prime? Shouldn't it be ##f'(x) = \frac{\cos x}{x}##?

songoku, Delta2 and berkeman
FactChecker said:
You forgot about the constant that is added to the integral. If you start the integral at p=3, then you know that the integral part is 0 at x=3. So what constant is added to the integral?
I understand

Mark44 said:
Did you forget to add the prime? Shouldn't it be ##f'(x) = \frac{\cos x}{x}##?
Yes, I am sorry

Thank you very much FactChecker and Mark44

## 1. How do I find f(x) from given f'(x)?

To find f(x) from given f'(x), you can use the integration method. This involves reversing the derivative process by finding the anti-derivative of f'(x). Once you have found the anti-derivative, you can add a constant term to get the general form of f(x).

## 2. What is the relationship between f(x) and f'(x)?

f(x) and f'(x) are related by the fundamental theorem of calculus, which states that the derivative of the integral of a function is equal to the original function. In other words, f'(x) is the derivative of f(x), and f(x) is the integral of f'(x).

## 3. Can I find f(x) if I only know f'(x)?

Yes, you can find f(x) if you only know f'(x). However, you will need to have some initial conditions or boundaries to solve for the constant term in the general form of f(x). Without these conditions, there will be an infinite number of possible solutions for f(x).

## 4. Are there any shortcuts or tricks for finding f(x) from given f'(x)?

There are some common integrals that you can memorize to make finding f(x) from given f'(x) faster. These include the power rule, product rule, quotient rule, and chain rule. However, for more complex functions, you will need to use integration techniques like substitution or integration by parts.

## 5. What happens if I make a mistake while finding f(x) from given f'(x)?

If you make a mistake while finding f(x) from given f'(x), you may end up with an incorrect solution. It is important to double-check your work and use multiple methods to verify your answer. Additionally, you can use online tools or graphing calculators to check your solution graphically.

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