How do I set this antiderivative up?

In summary: If you mean ##f(x) = \sqrt{x^3+x+1}## then ##F(a) = \int_0^a f(x) \, dx## evaluates to a horrible expression involving Elliptic functions. For the record, here is what I get when I do the integral using the computer algebra package Maple:.400000000000000*a*(a^3+a+1.)^(1/2)+747347077668721+.334696371293003e-1*(1944.39383604376-905.253833500497*a+1326.71397592448*a^2
  • #1
jakka
2
0
Mod note: Moved from technical math section, so there's no template.
Sorry if I'm formatting this question wrong, new user.
F(x) is an antiderivative of [PLAIN]https://upload.wikimedia.org/math/9/1/5/915ca58b070b0328cd069524c2d487f2.pngx[SUP]3[/SUP]+x+1. F(1)https://upload.wikimedia.org/math/4/3/e/43ec3e5dee6e706af7766fffea512721.png -2.125. Find F(4).
I tried to set up the antiderivative as (2/3)(x^3+x+1)^(3/2) and then plugging 1 into get -2.125, but its clearly wrong. Is there another step I need to do?
 
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  • #2
posted in the wrong place sorry thought this was the homework forum. how do i delete post?
 
  • #3
jakka said:
Sorry if I'm formatting this question wrong, new user.
F(x) is an antiderivative of [PLAIN]https://upload.wikimedia.org/math/9/1/5/915ca58b070b0328cd069524c2d487f2.pngx[SUP]3[/SUP]+x+1. F(1)https://upload.wikimedia.org/math/4/3/e/43ec3e5dee6e706af7766fffea512721.png -2.125. Find F(4).
I tried to set up the antiderivative as (2/3)(x^3+x+1)^(3/2)
No, that isn't right, assuming that the integrand is ##\sqrt{x^3 + x + 1}##. If you differentiate your proposed antiderivative, you don't get ##\sqrt{x^3 + x + 1}##.
jakka said:
and then plugging 1 into get -2.125, but its clearly wrong. Is there another step I need to do?

Are you sure you have copied the problem correctly? As written this is not a simple integration problem.
 
Last edited by a moderator:
  • #4
jakka said:
Mod note: Moved from technical math section, so there's no template.
Sorry if I'm formatting this question wrong, new user.
F(x) is an antiderivative of [PLAIN]https://upload.wikimedia.org/math/9/1/5/915ca58b070b0328cd069524c2d487f2.pngx[SUP]3[/SUP]+x+1. F(1)https://upload.wikimedia.org/math/4/3/e/43ec3e5dee6e706af7766fffea512721.png -2.125. Find F(4).
I tried to set up the antiderivative as (2/3)(x^3+x+1)^(3/2) and then plugging 1 into get -2.125, but its clearly wrong. Is there another step I need to do?

If you mean ##f(x) = \sqrt{x^3+x+1}## then ##F(a) = \int_0^a f(x) \, dx## evaluates to a horrible expression involving Elliptic functions. For the record, here is what I get when I do the integral using the computer algebra package Maple:
.400000000000000*a*(a^3+a+1.)^(1/2)+.747347077668721+.334696371293003e-1*(1944.39383604376-905.253833500497*a+1326.71397592448*a^2)^(1/2)/(6.33800237600924+9.28879394984575*a)^(1/2)-.760562326439714*EllipticF(29.7401409537626*(6.33800237600924+9.28879394984575*a)^(1/2)/(81.2424695605669+36.4240851075835*a),.911348956138490)+.995391725896672*EllipticE(29.7401409537626*(6.33800237600924+9.28879394984575*a)^(1/2)/(81.2424695605669+36.4240851075835*a),.911348956138490)+.283999288535776/(81.2424695605669+36.4240851075835*a)*(-.536375634160795e-1*(382.891652874454*a-221.118995987414*a^2-165.754662083937)*(2061.23266890537*a^2-1406.43636015277*a+3020.88330175229))^(1/2)/(6.33800237600924+9.28879394984575*a)^(1/2)
 
Last edited by a moderator:

1. What is an antiderivative?

An antiderivative is the inverse operation of a derivative. It is a function that, when differentiated, gives the original function. In other words, it is the reverse process of finding the slope of a curve.

2. Why do we need to set up antiderivatives?

Antiderivatives are useful in finding the original function when only the derivative is known. They are also used in integration, which is an important tool in solving various mathematical problems in fields such as physics, engineering, and economics.

3. How do I identify which antiderivative to use?

The process of finding an antiderivative involves using rules and formulas, such as the power rule, product rule, and chain rule. It is important to have a good understanding of these rules and know when to apply them in order to set up the correct antiderivative.

4. Can I use a calculator to set up an antiderivative?

While calculators can be helpful in solving antiderivatives, it is important to have a good understanding of the rules and concepts behind them. Additionally, not all antiderivatives can be solved using a calculator, so it is important to know how to set them up manually as well.

5. Are there any common mistakes to avoid when setting up an antiderivative?

One common mistake is forgetting to include the constant of integration when finding an antiderivative. Another mistake is applying the wrong rule or formula, which can lead to an incorrect solution. It is important to double check your work and practice setting up antiderivatives to avoid these mistakes.

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