# How to solve indefinite integral of $e^{t\left(\pi \:-r\right)}\left(-\frac{ds}{dt}\right)^{1-\frac{1}{n}}$?

I have the following integral: $$\int e^{t\left(\pi -r\right)}\left(-\frac{\mathrm{d}s}{\mathrm{d}t}\right)^{1-\frac{1}{n}}\mathrm{d}t.$$ I am not sure if this is solvable. I have tried integration by parts but do not know how to handle the term $$\left(-\frac{\mathrm{d}s}{\mathrm{d}t}\right)^{1-\frac{1}{n}}$$ due to the presence of the exponent $$1-\frac{1}{n}$$. Also, I can't see if Feynman Integration (if this becomes a definite integral with bounds from $$0$$ to infinity) would apply here. I would be grateful for any help resolving this.

• Could you tell what is $s$ ? Sep 14 '20 at 4:29
• $s=f(t)$ is a function of $t$. $ds/dt$ may also be written as $f'(t)$
– user809100
Sep 14 '20 at 7:13
• $\int \:e^{t\left(\pi -r\right)}\left(-\frac{ds}{dt}\right)^{1-\frac{1}{n}}dt$ is not an equation because there is no $=$ in it. Sep 14 '20 at 7:31
• No, I am just trying to evaluate the integral (e.g. $\int \sin(x) dx=-\cos(x)+C$).
– user809100
Sep 14 '20 at 7:35
• Where did you find it? If $s=f(t)$, can we know $f(t)$? have you tried power series? Sep 14 '20 at 23:42

The integral actually has a a simple solution:

$$\int e^{t(\pi-r)}\left(-\frac{ds}{dt}\right)^{1-1/n}dt=(-1)^{1-1/n}\int e^{t(\pi-r)}(s'(t))^{1-1/n}dt$$

This is because $$\frac{ds}{dt}$$ is in a derivative form, and can be represented without raising $$dt$$ to a power.

For example: if $$n=1$$, $$s(t)=t$$, and $$r=\pi-1$$, then:

$$\int e^{t(\pi-r)}\left(-\frac{ds}{dt}\right)^{1-1/n}dt=\int e^tdt=e^t+C$$

• Thank you for your response. However, I am looking for a general form of the integral (I don't know values for $n$, $s(t)$, and $r$).
– user809100
Sep 16 '20 at 6:42
• @UNOWen The integral does not generally have an elementary form. Sep 16 '20 at 19:23

Since $$\;\dfrac{\text ds}{\text dt} = \left(\dfrac{\text dt}{\text ds}\right)^{-1},$$ then the given integral can be presented via the inverse function $$\;t(s):$$ $$\int e^{t(\pi-r)}\left(-\dfrac{\text ds}{\text dt}\right)^{\large 1-^1/_n}\,\text dt =-\int e^{(\pi-r)\,t(s)}\;\sqrt[\Large n]{-\dfrac{\text dt}{\text ds}}\,\text ds.$$

However, detalization of $$\;s(t)\;$$ or $$\;t(s)\;$$ looks nesessary.