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

[Jwill]

What method of integration would I use to find the anti-derivative of

$$\int sin(3u(t)) dt$$

when

$$u(t)$$

Is and unknown function of time?

 

[Whitishcube]

It depends, because there are many elementary functions that U could be that would make this a nonelementary integral.

 

[Jwill]

I mean if

$$u(t)$$

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

$$u'(t)$$?


$$\frac{-1}{3u'(t)}cos(3u(t))$$
seems to be sort of close if you differentiate it... But this is an incorrect usage of U substitution.

 

[paulfr]

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

 

[Jwill]

What do you mean by error functions? Could you give me an example of such a situation? I am not exact familiar with that.

 

[paulfr]

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.

 

[Jwill]

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.