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Here is the calculus way:
I have no prior knowledge of calculus. But my teacher asks me to derive the integrated rate law for any order between one to ten.
It would be fine for me to memorize the derivations for zero to third orders. But I can't do all ten for sure.
Calculus is pretty much just a shorthand for algebra (plus the whole limits concept), so the answer is technically yes.
Your teacher might be intending for you to use the pattern @Klaus Warzecha mentioned:
For a function $f(x) = x^n$, the primitive integral is $F(x) = \vert\ln(x)\vert$ if $n=−1$, otherwise $F(x) = \frac{1}{n+1}x^{n+1}$
So in that sense, you can use algebra if you let
$$\int_{A_0}^{A_\tau} \! \frac{1}{[A]^n} \, \mathrm{d}[A] = \frac{1}{n - 1} \left[ \frac{1}{[A]^{n-1}_{\tau}} - \frac{1}{[A]^{n-1}_{0}} \right]$$
for $n \neq 1$
and
$$\int_{A_0}^{A_\tau} \! \frac{1}{[A]} \, \mathrm{d}[A] = \ln \frac{[A]_\tau}{[A]_0}$$
for $n = 1$
Note that this only works for single-species rate equations - if you have multiple species the integration result changes. Note that this also isn't really using algebra - it's memorizing and applying one of many general integration results. Which is what you do in integral calculus.
In any case, it doesn't seem fair to me to expect you to know how to derive integrated rate laws if integral calculus isn't a prerequisite for your course. If it is a prereq, then now you are seeing why. If it isn't, then what your teacher is asking you to do is, in my opinion, unrealistic.
