Determine all primitive functions

• beyondlight
In summary: So you need to find a substitution that will allow you to use the chain rule. In this problem, the function inside the parentheses looks suspiciously like the derivative of part of the function you're integrating, so let's try using that as our substitution.Let z = x^2 + 3. Then dz/dx = 2x, or x = (1/2)dz. Make that substitution, and your integral will become(1/2)∫2(z)^4 dz = ∫z^4 dz. That's a pretty easy integral to do, and then you can substitute back in for z to get the answer in terms of x.
beyondlight

Homework Statement

Determine all primitive functions for the function:

2x(x^2+3)^4

2. The attempt at a solution

When i expanded i got the primitive to be:

2(x^9/9)+3x^8+18x^6+54x^4+81x^2But this was wrong. I am not sure I have understood the question. Help?

beyondlight said:

Homework Statement

Determine all primitive functions for the function:

2x(x^2+3)^4

2. The attempt at a solution

When i expanded i got the primitive to be:

2(x^9/9)+3x^8+18x^6+54x^4+81x^2But this was wrong. I am not sure I have understood the question. Help?
I have moved this thread, as it is not a precalculus problem.

Apparently the problem asks you to find all antiderivatives of 2x(x2 + 3)4. In other words, carry out this integration: ##\int 2x(x^2 + 3)^4 dx##. The answer should have a highest degree term of x10. Don't forget the constant of integration.
Instead of expanding the binomial in parentheses, think about a simple substitution that you can do.

One possible substitution is 2x(z)^4

But I am not sure how to proceed from here...

beyondlight said:
One possible substitution is 2x(z)^4
If z = x^2 + 3, what is dz?
beyondlight said:
But I am not sure how to proceed from here...
A very simple substitution will work, and you're on the right track,
When you use substitution to evaluate an integral, you're using the chain rule in reverse.

1. What is a primitive function?

A primitive function is a fundamental function that cannot be further simplified or decomposed into simpler functions. It is the most basic form of a function and is often used as a starting point for more complex functions.

2. How do you determine all primitive functions?

To determine all primitive functions, you need to use the process of integration. This involves finding the antiderivative of a given function, which is the function that, when differentiated, gives the original function. The set of all antiderivatives is equivalent to the set of all primitive functions.

3. Can all functions have primitive functions?

No, not all functions have primitive functions. Only continuous functions have primitive functions. Discontinuous functions or functions with discontinuities at certain points do not have primitive functions.

4. How do you find the primitive function of a specific function?

To find the primitive function of a specific function, you can use techniques such as integration by parts, substitution, or partial fractions. These techniques help you manipulate the given function to find its antiderivative, which is the primitive function.

5. Why are primitive functions important?

Primitive functions are important because they allow us to solve complex problems involving derivatives and integrals. They also help us understand the behavior and properties of functions. In addition, primitive functions have many practical applications in fields such as physics, engineering, and economics.

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