Antiderivatives (find f for )

In summary, antiderivatives are mathematical functions that are the reverse operation of derivatives and are used to find the indefinite integral of a given function. To find an antiderivative, one must use the rules of integration and add a constant term. However, not all functions have an antiderivative, and some can only be integrated using non-elementary functions. Antiderivatives are also related to finding the area under a curve, as the definite integral is the reverse operation of finding the antiderivative. In real-world applications, antiderivatives are used in various fields, such as physics, engineering, and economics, to model and analyze real-world problems.
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Antiderivatives - 8x - 3 sec^2x

Im not sure whether my answer to this antiderivative question is correct and would like your opinion.

1. The antiderivative statement asks: Find f

2. f prime (x) = 8x - 3 sec^2x
Most general antiderivative = F(x) + C
Antidefferentiation formula: Function = sec^2, Particular antiderivative = tan x

3. My attempt at a solution
= [(8x^2)/2] - 3tan x
= 4x^2 - 3tan x

Is this correct?
 
Last edited:
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  • #2
looks good to me, nice job.
 
  • #3
physicsman2 said:
looks good to me, nice job.

Ok, great, thanks.
 

1. What are antiderivatives?

Antiderivatives, also known as indefinite integrals, are the reverse operation of derivatives. They are mathematical functions that, when differentiated, yield a given function.

2. How do you find the antiderivative of a function?

To find the antiderivative of a function, use the rules of integration to manipulate the function and add a constant term. The constant is known as the constant of integration and accounts for all possible antiderivatives of the function.

3. Can all functions have an antiderivative?

No, not all functions have an antiderivative. If a function is not continuous or does not have a finite number of discontinuities, it does not have an antiderivative. Additionally, some functions cannot be integrated using elementary functions.

4. What is the relationship between antiderivatives and area under a curve?

The definite integral, which involves finding the area under a curve, is the reverse operation of finding the antiderivative. This means that the definite integral of a function is equal to the difference between the antiderivatives of the function evaluated at the upper and lower bounds of integration.

5. How are antiderivatives used in real-world applications?

Antiderivatives are used in various real-world applications, such as calculating work done by a force, finding the center of mass of an object, and calculating the displacement and velocity of an object. They are also used in economics, physics, and engineering to model and analyze real-world problems.

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