Finding Anti-Derivative of sin(3u(t)) w/ Unknown u(t)

In summary, the method of integration used to find the anti-derivative of \int sin(3u(t)) dt when u(t) is an unknown function of time depends on the complexity of u(t). If u(t) is a linear function, the antiderivative can be found using basic integration techniques. However, if u(t) is more complex, the integral will involve error functions and cannot be integrated numerically. Examples of such situations can be found by using WolframAlpha to calculate integrals and exploring the use of error functions.
  • #1
Jwill
39
0
What method of integration would I use to find the anti-derivative of

[tex]\int sin(3u(t)) dt[/tex]

when

[tex]u(t)[/tex]

Is and unknown function of time?
 
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  • #2
It depends, because there are many elementary functions that U could be that would make this a nonelementary integral.
 
  • #3
I mean if

[tex]u(t)[/tex]

is completely unknown, is there no way to just generally integrate it even using terms like

[tex]u'(t)[/tex]?


[tex]\frac{-1}{3u'(t)}cos(3u(t))[/tex]
seems to be sort of close if you differentiate it... But this is an incorrect usage of U substitution.
 
  • #4
If u(t) is unknown, you can not integrate numerically, and in general
if u(t) is anything more complex than a linear function in t, the integral
will involve error functions.

For example, if u(t) = t, your antiderivative in post # 3 would be correct.
Similarly if u(t) = kt +b with k and b constants
 
  • #5
What do you mean by error functions? Could you give me an example of such a situation? I am not exact familiar with that.
 
  • #6
Jwill said:
What do you mean by error functions? Could you give me an example of such a situation? I am not exact familiar with that.

Go to www.wolframAlpha.com

Type in " integral e^ (3x^2) dx " [omit quotes] and hit return.
See the erf error function in the answer ?

Then clear and

Type in erf x and hit return
Put the cursor over the lower right "erf is the error function"
and click on definitionNote that if you change 3x^2 to 3x, the error function is not needed.
 
  • #7
Okay, thanks. I know what you mean now... I actually feel kinda stupid lol. I've actually found exact error from numerical solutions of different orders.
 

1. What is an anti-derivative?

An anti-derivative is the opposite operation of taking a derivative. It is a function that, when differentiated, gives the original function.

2. How do I find the anti-derivative of sin(3u(t))?

To find the anti-derivative of sin(3u(t)), you can use the substitution method. Let u(t) be equal to a variable, such as x, and then use the chain rule to find the anti-derivative.

3. What is the purpose of using the substitution method to find the anti-derivative?

The substitution method allows us to simplify the function and make it easier to integrate. It also helps to identify patterns and solve more complex integrals.

4. How do I handle the unknown variable u(t) when finding the anti-derivative?

When finding the anti-derivative of a function with an unknown variable, such as u(t), treat it as a constant and integrate as usual. Then, substitute the original variable back in at the end.

5. Are there any other methods for finding the anti-derivative of sin(3u(t))?

Yes, you can also use integration by parts or trigonometric identities to find the anti-derivative. However, the substitution method is often the most straightforward and efficient for this particular function.

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