Indefinite integral for the square root of a quartic function

In summary, the problem involves finding the length of a 3D cubic interpolation using the formula ((At4+Bt3+Ct2+Dt+E)1/2)dt, where A, B, C, D, and E are constants that need to be plugged in after integration. This requires the use of reduction formulas, but since the function is quartic, it may need to be converted into a quadratic function. However, this may not be possible and the integral involves the inverse sine function and elliptic integrals of the first and third kinds. Wolfram-Alpha cannot evaluate the integral, but a helpful website has been provided.
  • #1
StTheo
2
0
I arrived at this problem while trying to find the length of a 3D http://en.wikipedia.org/wiki/Cubic_interpolation" .

Basically, I'm having to figure out how to integrate ((At4+Bt3+Ct2+Dt+E)1/2)dt

A,B,C,D, and E are constants which have to be plugged in after integration, I'm sorry to say.

All the reduction formulas I've seen use quadratic functions of some kind, so would I have to convert the quartic function into some kind of quadratic function?
 
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  • #2
In general, it cannot be done simply, I believe. From what I can tell using Wolfram-Alpha, the integral involves the inverse sine function as well as the elliptic integrals of the first AND third kinds. It can't evaluate the integral, though: here.
 
  • #3
Grr, oh well. However, I hadn't seen that wonderful site you linked to before. I'll probably waste quite a bit of time on it.
 

1. What is the definition of an indefinite integral?

An indefinite integral is a mathematical operation that involves finding the antiderivative of a function. In other words, it is the reverse process of taking a derivative, and it helps us to find the original function from its derivative.

2. How do you find the indefinite integral of a square root function?

To find the indefinite integral of a square root function, we first need to rewrite the function in the form of x^n, where n is a positive integer. For a square root function, we can rewrite it as x^(1/2). Then, we apply the power rule, which states that the integral of x^n is (x^(n+1))/(n+1) + C. Therefore, the indefinite integral of the square root function is (x^(3/2))/(3/2) + C, which simplifies to (2/3)x^(3/2) + C.

3. What is the relationship between indefinite integrals and definite integrals?

The indefinite integral of a function is the general solution, while the definite integral is a specific solution. The definite integral involves evaluating the indefinite integral at specific limits of integration, which gives us a numerical value. In other words, the definite integral is the area under the curve of the function within a specific interval.

4. Can you find the indefinite integral of a quartic function?

Yes, the indefinite integral of a quartic function can be found using the power rule mentioned earlier. We need to rewrite the quartic function in the form of x^n, where n is a positive integer, and then apply the power rule to find the antiderivative. The resulting indefinite integral will be a polynomial function.

5. Why is the indefinite integral of a quartic function important in science?

The indefinite integral of a quartic function is important in science because it helps us to model and understand various natural phenomena. Quartic functions can represent many real-world scenarios, such as the motion of objects under constant acceleration, population growth, and radioactive decay. By finding the indefinite integral, we can determine the original function and make predictions about how the phenomenon will change over time.

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