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\title{Math 242, Lecture 3}
\maketitle

\subsection{ A first application of integration}

\begin{example}
Basic physics dictates that, neglecting air speed,
the downward acceleration of an object dropped
due to gravity is constant, $32$ feet per second per second.   
Find a formula for the distance travelled by an object dropped (not thrown,
so it starts with no speed) over $t$ seconds.     How long would it take
an object to fall 5000 feet (for example dropping it over the edge of the Grand
Canyon)?  
\end{example}

This  example marks the beginning of the subject of {\em differential 
equations},   which is the study of how to solve equations 
involving derivatives.     All of the laws of basic, classical
physics are written in terms of differential equations.  

\section{ Techniques of integration}

So far, our method for finding anti-derivatives has 
been to guess, check (by taking the derivative
of our guess) and modify until the answer is correct.    It would
be better to have a more systematic approach.  Unfortunately, while differentiation
requires only a few rules to take the derivatives of most familiar functions,
anti-differentiation is a much more complicated game, involving many more
rules (so many that at MIT they hold an ``Integration Bee'').   Moreover,
there are many functions, such as $e^{x^2}$,  
which have no simple anti-derivative.  

The first way to be systematic about integration is to find analogues of the
first rules we used for differentiation.  
For example, if $F(x)$ is an anti-derivative for $f(x)$   - so that
$\frac{d}{dx} F(x) = f(x)$,    and similarly $\frac{d}{dx} G(x) = g(x)$,  
then $\frac{d}{dx} [F(x) + G(x)] =   f(x) + g(x)$.   So that
$F(x) + G(x)$ is an anti-derivative for $f(x) + g(x)$.  
This is the first of the rules listed below.    All of these rules
may be checked by differentiating.  

\subsection{ First integration rules}

\begin{itemize}
\item $\int(f(x) + g(x)) dx = \int f(x) dx + \int g(x) dx$.  
\item For any constant number $c$, $\int c f(x) dx = c \int f(x) dx$.  
\item $\int x^n dx = \frac{1}{n+1}x^{n+1} + C$ if $n \neq -1$.  
\item $\int e^{kx} dx = \frac{1}{k} e^{kx} + C$.  
\item $\int \frac{1}{x} dx = \ln |x| + C$.  
\end{itemize}

Looking at the first two rules, do you think there will be an easy rule
involving $\int f(x) \cdot g(x) dx$?  

\begin{example}
Evaluate the following indefinite integrals:
\begin{itemize}
\item $\int (2x^3 - 5x + 3) dx$.  
\item $\int e^{3x}  dx$.  
\item $\int [\frac{5}{x} + \frac{3}{x^2}]  dx$.  
\end{itemize}
\end{example}

 

\subsection{ Integration by substitution} 

While we might not have thought about it, 
the chain rule was by far the rule we used most
frequently in finding derivatives.    
In the world of integration, the cousin of the
chain rule is the technique of ``u-substitution''
for integration.     We start with a couple examples
which we attempt before formalizing the process.  

\begin{example}
Evaluate $\int (2x - 3)^{10} 20 dx$.   
\end{example}

\begin{example}
Evaluate $\int (x^2 - 1)^{5} (10 x )dx$   and  $\int x (x^2 - 1)^{5}  dx$   
\end{example}

We make this systematic through steps similar to 
the steps we took when applying the chain rule.  
To evaluate $\int f(x) dx$, 
\begin{itemize}
\item First we identify a function traditionally called
$u(x)$   (hence the name ``u-substitution'')   
which we use as a building block to make the integrand.  
In particular, we want to have some $g(u(x))$ appearing
naturally as part of the integrand.  
\item Then, we take the derivative of $u(x)$,  
which is part of the chain rule.  
\item We substitute the variable $u$ in for $u(x)$   (this is
easy)   and then - here's the rub! -   try to match what's remaining with
$du$   (which is $\frac{du}{dx} dx$).    In this step we often use 
the trick of multiplying by some number $k$ inside the integral and
then $\frac{1}{k}$ outside the integral.  
\item If the substitution meshes, we have successfully translated the
integral $\int f(x) dx = \int g(u) du$.  
\item If we can integrate $\int g(u) du = G(u) + C$   then
we can substitute $u(x)$ for $u$ to find the original integral.  
\end{itemize}

The only way to learn this technique is through  plenty of practice.  
(It's a bit of an art).  

\begin{example}
Evaluate the following integrals:
\begin{itemize}
\item $\int (x^4 + 2x - 1)^{-5} (2x^3 + 1) dx$  
\item $\int e^{x^3 - 1} x^2 dx$  
\item $\int \frac{1}{x^2 - 4} x dx$  
\item $\int \frac{1}{x \ln x} dx$  
\item $\int \frac{x^2 + 2}{x + 1} dx$  
\item $\int x^2 \sqrt{x^3 + 5} dx$ 
\item $\int \frac{e^x}{e^x + 1} dx$  
\item $\int e^{-x^2} dx$  
\end{itemize}

\end{example}









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