Integrals appearing in Kinetic Problems

Since a kinetic system is defined by a system of differential equations, the solutions in the simplest cases are obtained by integration of a separable equation, whereas the more complex systems can be solved as Riccati, Bernoulli and Logistic equations, for example. Below is a list of solutions arising from several common separable differential equations which appear in kinetic problems.

First-order kinetics: Logarithmic solution

`int 1/xdx = ln x`

Second and Third-order kinetics: Polynomial integral

`int x^n dx = 1/(n+1)x^(n+1)`

First-and-a-half order kinetics: Arctangent anti-derivative and Hyperbolic Arctangent

`int 1/(1+x^2)dx = arctan x`

`int 1/(1-x^2)dx = "arctanh " x`

Parallel-consecutive bimolecular kinetics: Lerch-transcendent

`int (x^(a-1))/(1-zx) dx = Phi(z,1,a)`

Hypergeometric: Various complex schemes

`int (x^(f-1)(1-x)^(g-f-1))/(1-zx)^e dx = (Gamma(g-f)Gamma(f))/(Gamma(g)) " "_2F_1(e,f,g;z)`

`int = Ei(x)`