Antiderivative of (x^2 + 4x)^(1/3)

In summary, the conversation discusses finding the value of g(1) for a given function f, where g is an antiderivative of f and g(5) = 7. It is suggested to find the antiderivative of f and add a constant, choosing the value of C to make g(5) = 7. Then, g(1) is found using the antiderivative and the chosen value of C. It is also mentioned that there may be a way to represent g(1) in terms of g(5) and a definite integral of f, but it may require numerical integration.
  • #1
jesuslovesu
198
0
If f is the function defined by f(x) = (x^2 + 4x)^(1/3) and g is an antiderivative of f such that g(5) = 7 then g(1) =

I thought that I need to find the antiderivative of f but it turns out that it's really messy so I'm not sure, is there something I'm missing to be able to solve for g(1)?
 
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  • #2
You were on the right track. Find the anti derivative G, and add a constant, C. (Do you know why?) Now choose C such that G(5) = 7. Then find G(1) using the antiderivative plus the C you just found.
 
  • #3
You already KNOW that g is an anti-derivative of f!
Now, how can you therefore represent g(1) in terms of g(5) and a definite integral of f?
 
  • #4
Do you want a numerical answer or would g(1)= a formula be sufficient? I think that is what arildno is saying.
 
  • #5
Agreed:
I can't see any obvious way to evaluate the definite integral other than through numerical integration.
 

What is an antiderivative?

An antiderivative is the reverse of a derivative. It is a function that, when differentiated, gives the original function.

What is the antiderivative of (x^2 + 4x)^(1/3)?

The antiderivative of (x^2 + 4x)^(1/3) is (3/5)(x^2 + 4x)^(5/3) + C, where C is a constant.

How do you find the antiderivative of (x^2 + 4x)^(1/3)?

To find the antiderivative of (x^2 + 4x)^(1/3), we can use the power rule for integrals and then simplify the resulting expression.

What is the general formula for finding an antiderivative?

The general formula for finding an antiderivative is to reverse the power rule for derivatives. This means that for any function f(x), the antiderivative is F(x) = ∫f(x)dx + C, where C is a constant.

Can the antiderivative of a function have multiple solutions?

Yes, the antiderivative of a function can have multiple solutions because a constant can be added to the antiderivative without changing its derivative. This means that there can be an infinite number of antiderivatives for a given function.

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