Determine all primitive functions

[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?

 

[Mark44]

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.

 

[beyondlight]

One possible substitution is 2x(z)^4

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

 

[Mark44]

One possible substitution is 2x(z)^4

If z = x^2 + 3, what is dz?

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.