I cannot find any sources that actually list the steps for integrating a rate law, and it’s been driving me crazy. My Ap chem 2 teacher doesn’t know how to do it and just wants me to memorize the common rate laws, however I don’t memorize equations unless I know how to write them out and where they come from. I have the equation
$$ \text{rate} = \dfrac{\mathrm d[A]}{\mathrm dt} = k \cdot [A_0]^{n} $$
where $n$ is the order of the equation
$[A_0]$ is the concentration at time $0$
$[A]$ is the concentration with respect to time
$k$ is the rate constant
I know that for a first order ($n=1$) reaction the equation becomes
$$ [A] = [A_0] \cdot \mathrm e^{kt} $$
and that a second order ($n=2$) reaction the equation is
$$ [A] = \dfrac{[A_0]}{1 + kt \cdot [A_0]} $$
But I don't know what the steps are to find these equations. If I find that the order of my equation needs to be 1.5 ($n=1.5$), I will have no Idea what the equation will be.
From these equations I will know most of the time $[A_0]$, $[A]$ and sometimes Rate.
Edit:
After following the links I learned a portion of the process to solve the equation for any $n$. However after testing it out I am either getting a incorrect answer of a incomplete answer
$$ [A] =\int_{[A_0]}^{[A]}{\frac{1}{[A_0]^2}\,\mathrm d[A]} = \frac{[A_0]^{1}}{1} = \int_{0}^{t}{k\,\mathrm dt}$$
the answer should be
$$ [A] = \dfrac{[A_0]}{1 + kt \cdot [A_0]} $$
what parts am I missing?