CME 102 - Ordinary Differential EquationsCalculus refresher
By Afshine Amidi and Shervine Amidi
Integral calculus
Primitive function The primitive function of a function $f$, noted $F$ and also known as an antiderivative, is a differentiable function such that:
Integral Given a function $f$ and an interval $[a,b]$, the integral of $f$ over $[a,b]$, noted $\int_a^bf(x)dx$, is the signed area of the region in the $xy$-plane that is bounded by the graph of $f$, the $x$-axis and the vertical lines $x=a$ and $x=b$, and can be computed with the primitive of $f$ as follows:
Integration by parts Given two functions $f, g$ on the interval $[a,b]$, we can integrate by parts the quantity $\int_a^bf(x)g'(x)dx$ as follows:
Rational primitive functions The table below sums up the main rational functions associated to their primitives. We will omit the additive constant $C$ associated to all those primitives.
| Function $f$ | Primitive $F$ |
| $\displaystyle a$ | $\displaystyle ax$ |
| $\displaystyle x^a$ | $\displaystyle \frac{x^{a+1}}{a+1}$ |
| $\displaystyle \frac{1}{x}$ | $\displaystyle \ln|x|$ |
| $\displaystyle \frac{1}{1+x^2}$ | $\displaystyle \textrm{arctan}(x)$ |
| $\displaystyle \frac{1}{1-x^2}$ | $\displaystyle \frac{1}{2}\ln\left|\frac{x+1}{x-1}\right|$ |
Irrational primitive functions The table below sums up the main rational functions associated to their primitives. We will omit the additive constant $C$ associated to all those primitives.
| Function $f$ | Primitive $F$ |
| $\displaystyle \frac{1}{\sqrt{1-x^2}}$ | $\displaystyle \textrm{arcsin}(x)$ |
| $\displaystyle -\frac{1}{\sqrt{1-x^2}}$ | $\displaystyle \textrm{arccos}(x)$ |
| $\displaystyle \frac{x}{\sqrt{x^2-1}}$ | $\displaystyle \sqrt{x^2-1}$ |
Exponential primitive functions The table below sums up the main exponential functions associated to their primitives. We will omit the additive constant $C$ associated to all those primitives.
| Function $f$ | Primitive $F$ |
| $\displaystyle \ln(x)$ | $\displaystyle x\ln(x)-x$ |
| $\displaystyle \exp(x)$ | $\displaystyle \exp(x)$ |
Trigonometric primitive functions The table below sums up the main trigonometric functions associated to their primitives. We will omit the additive constant $C$ associated to all those primitives.
| Function $f$ | Primitive $F$ |
| $\displaystyle \cos(x)$ | $\displaystyle \sin(x)$ |
| $\displaystyle \sin(x)$ | $\displaystyle -\cos(x)$ |
| $\displaystyle \tan(x)$ | $\displaystyle -\ln|\cos(x)|$ |
| $\displaystyle \frac{1}{\cos(x)}$ | $\displaystyle \ln\left|\tan\left(\frac{x}{2}+\frac{\pi}{4}\right)\right|$ |
| $\displaystyle \frac{1}{\sin(x)}$ | $\displaystyle \ln\left|\tan\left(\frac{x}{2}\right)\right|$ |
| $\displaystyle \frac{1}{\tan(x)}$ | $\displaystyle \ln|\sin(x)|$ |
Laplace transforms
Definition The Laplace transform of a given function $f$ defined for all $t\geqslant0$ is noted $\mathscr{L}(f)$, and is defined as:
Remark: we note that $f(t)=\mathscr{L}^{-1}(F)$ where $\mathscr{L}^{-1}$ is the inverse Laplace transform.
Main properties The table below sums up the main properties of the Laplace transform:
| Property | $t$-domain | $s$-domain |
| Linearity | $\alpha f(t)+\beta g(t)$ | $\alpha F(s)+\beta G(s)$ |
| $t$-domain integral | $\displaystyle\int_0^tf(\tau)d\tau$ | $\displaystyle\frac{F(s)}{s}$ |
| $t$-domain first derivative | $f'(t)$ | $sF(s)-f(0)$ |
| $t$-domain second derivative | $f''(t)$ | $s^2F(s)-sf(0)-f'(0)$ |
| $t$-domain $n^{th}$ derivative | $f^{(n)}(t)$ | $s^nF(s)-s^{n-1}f(0)-...-sf^{(n-2)}(0)-f^{(n-1)}(0)$ |
| $s$-domain integral | $\displaystyle\frac{f(t)}{t}$ | $\displaystyle\int_s^{+\infty}F(\sigma)d\sigma$ |
| $s$-domain first derivative | $tf(t)$ | $-F'(s)$ |
| $s$-domain second derivative | $t^2f(t)$ | $F''(s)$ |
| $s$-domain $n^{th}$ derivative | $t^nf(t)$ | $(-1)^nF^{(n)}(s)$ |
Common transform pairs The table below sums up the most common Laplace transform pairs:
| $t$-domain | $s$-domain |
| $a$ | $\displaystyle\frac{a}{s}$ |
| $t$ | $\displaystyle\frac{1}{s^2}$ |
| $t^n$ | $\displaystyle\frac{n!}{s^{n+1}}$ |
| $e^{at}$ | $\displaystyle\frac{1}{s-a}$ |
| $\cos(\omega t)$ | $\displaystyle\frac{s}{s^2+\omega^2}$ |
| $\sin(\omega t)$ | $\displaystyle\frac{\omega}{s^2+\omega^2}$ |
| $\textrm{cosh}(at)$ | $\displaystyle\frac{s}{s^2-a^2}$ |
| $\textrm{sinh}(at)$ | $\displaystyle\frac{a}{s^2-a^2}$ |
Main operations The table below sums up the main operations of the Laplace transform:
| Operation | $t$-domain | $s$-domain |
| Unit step function | $u(t-a)$ | $\frac{e^{-as}}{s}$ |
| Dirac delta function | $\delta(t-a)$ | $e^{-as}$ |
| $s$-shift | $e^{at}f(t)$ | $F(s-a)$ |
| $t$-shift | $u(t-a)f(t-a)$ | $e^{-as}F(s)$ |