What to do when there's lot of function of time? (For integration) Should I consider them as constant?
Let I have a function which is looking like this
$$f=\int\frac{l \ \mathrm {dt}}{\dot{\theta}r\dot{r}}$$
Here $\dot{\theta}$, $r$, $\dot{r}$ and $l$ all of them are function of $t$. For differentiating respect to time I simply increase "dot". But what to do for integration? Should I consider them as constant?
What will happen if I forcefully say that $r$ isn't constant? $r$ is changing every single sec.
1 answer
No, if you're integrating with respect to time, you can't treat functions of time as constants.
Remember that, by the fundamental theorem of calculus, the result of your integration must be a function that, when differentiated, yields your original integrand. So if you incorrectly concluded that \(\int \frac{l}{\dot\theta r \dot r}dt = \frac{l}{\dot\theta r \dot r}t\;(+\;C)\), you would need to be able to show that \(\frac{d}{dt}\frac{l}{\dot\theta r \dot r}t = \frac{l}{\dot\theta r \dot r}\). By the product rule, \(\frac{d}{dt}\frac{l}{\dot\theta r \dot r}t = \frac{l}{\dot\theta r \dot r} + t\frac{d}{dt}\frac{l}{\dot\theta r \dot r}\), so that's the right answer only if \(\frac{d}{dt}\frac{l}{\dot\theta r \dot r} = 0\)—in other words, only if those functions combined in that way really is a constant.
To correctly solve the above integral, you'd need more information about those functions—if you know the definitions of the functions, you could substitute those in and then (maybe!) the integral will be solvable.
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